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<Step 1> Q - Assumption <Step 2> Implies(Q, Q) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 11> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Deducted <Step 12> Step 10 and Step 11 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))) - Modus Ponens <Step 13> Implies(P, Implies(Q, P)) - Axiom 1 <Step 14> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))) - Deducted <Step 15> Step 13 and Step 14 -> Implies(P, Implies(P, Q)) - Modus Ponens <Step 16> Implies(P, Implies(P, Q)) - Deducted

<Goal> Implies(P, Implies(P, Q)) <Assumption> Q <Assumption> Implies(Q, Q) <Assumption> P <Assumption> Implies(P, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Axiom 1 <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 10> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 11> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted <Step 12> Step 10 and Step 11 -> P - Modus Ponens <Step 13> P - Deducted

<Goal> P <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Assumption <Step 3> Implies(P, Implies(Q, P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Q)))) - Assumption <Step 5> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Q))) - Modus Ponens <Step 7> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted <Step 9> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Q))) - Deducted <Step 11> Step 7 and Step 8 -> P - Modus Ponens <Step 12> Step 9 and Step 10 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Q)) - Modus Ponens <Step 13> P - Deducted <Step 14> Implies(P, Implies(Implies(P, Implies(Q, P)), Q)) - Deducted <Step 15> Step 13 and Step 14 -> Implies(Implies(P, Implies(Q, P)), Q) - Modus Ponens <Step 16> Implies(P, Implies(Q, P)) - Axiom 1 <Step 17> Implies(Implies(P, Implies(Q, P)), Q) - Deducted <Step 18> Step 16 and Step 17 -> Q - Modus Ponens <Step 19> Q - Deducted

<Goal> Q <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Q))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Axiom 1 <Step 3> P - Assumption <Step 4> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, P))) - Assumption <Step 5> Implies(P, Implies(Q, P)) - Axiom 1 <Step 6> Implies(Implies(P, Implies(Q, P)), P) - Assumption <Step 7> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))))) - Assumption <Step 9> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 10> Step 3 and Step 4 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Modus Ponens <Step 11> Step 5 and Step 6 -> P - Modus Ponens <Step 12> Step 7 and Step 8 -> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Modus Ponens <Step 13> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 14> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 15> Implies(P, Implies(Q, P)) - Axiom 1 <Step 16> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Deducted <Step 17> P - Deducted <Step 18> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 19> Step 13 and Step 14 -> Q - Modus Ponens <Step 20> Step 15 and Step 16 -> Implies(Q, P) - Modus Ponens <Step 21> Step 17 and Step 18 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 22> Q - Deducted <Step 23> Implies(Q, P) - Deducted <Step 24> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 25> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted <Step 26> Step 22 and Step 23 -> P - Modus Ponens <Step 27> Step 24 and Step 25 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Modus Ponens <Step 28> P - Deducted <Step 29> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Deducted <Step 30> Step 28 and Step 29 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) - Modus Ponens <Step 31> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) <Assumption> Q <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, P))) <Assumption> Implies(Implies(P, Implies(Q, P)), P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), P) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, P) - Assumption <Step 5> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 6> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))))) - Assumption <Step 7> Step 1 and Step 2 -> P - Modus Ponens <Step 8> Step 3 and Step 4 -> P - Modus Ponens <Step 9> Step 5 and Step 6 -> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Modus Ponens <Step 10> P - Deducted <Step 11> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Axiom 1 <Step 12> P - Deducted <Step 13> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Deducted <Step 14> Step 10 and Step 11 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 15> Step 12 and Step 13 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Modus Ponens <Step 16> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 17> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted <Step 18> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 19> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Deducted <Step 20> Step 16 and Step 17 -> P - Modus Ponens <Step 21> Step 18 and Step 19 -> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Modus Ponens <Step 22> P - Deducted <Step 23> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Deducted <Step 24> Step 22 and Step 23 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Modus Ponens <Step 25> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) <Assumption> Implies(Implies(P, Implies(Q, P)), P) <Assumption> P <Assumption> Implies(P, P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 7> Implies(P, Implies(Q, P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q)))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, Implies(P, Implies(P, Q))) - Modus Ponens <Step 10> Implies(P, Implies(P, Implies(P, Q))) - Deducted

<Goal> Implies(P, Implies(P, Implies(P, Q))) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))))

<Step 1> P - Assumption <Step 2> Implies(P, Q) - Assumption <Step 3> Implies(P, Implies(Q, P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), P))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), P) - Modus Ponens <Step 10> Implies(P, Implies(Q, P)) - Axiom 1 <Step 11> Implies(Implies(P, Implies(Q, P)), P) - Deducted <Step 12> Step 10 and Step 11 -> P - Modus Ponens <Step 13> P - Deducted

<Goal> P <Assumption> P <Assumption> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), P)))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), P) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 11> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q)))) - Deducted <Step 12> Step 10 and Step 11 -> Implies(Q, Implies(P, Implies(P, Q))) - Modus Ponens <Step 13> Implies(Q, Implies(P, Implies(P, Q))) - Deducted

<Goal> Implies(Q, Implies(P, Implies(P, Q))) <Assumption> Implies(Implies(P, Implies(Q, P)), P) <Assumption> Q <Assumption> Implies(Q, Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Q) - Modus Ponens <Step 7> Implies(P, Implies(Q, P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Q, P)), Q) - Deducted <Step 9> Step 7 and Step 8 -> Q - Modus Ponens <Step 10> Q - Deducted

<Goal> Q <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) - Assumption <Step 5> Q - Assumption <Step 6> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 7> Step 1 and Step 2 -> Q - Modus Ponens <Step 8> Step 3 and Step 4 -> Implies(Q, P) - Modus Ponens <Step 9> Step 5 and Step 6 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 10> Q - Deducted <Step 11> Implies(Q, P) - Deducted <Step 12> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 13> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Deducted <Step 14> Step 10 and Step 11 -> P - Modus Ponens <Step 15> Step 12 and Step 13 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 16> P - Deducted <Step 17> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Deducted <Step 18> Step 16 and Step 17 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))) - Modus Ponens <Step 19> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 7> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, P)))) - Assumption <Step 5> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, P))) - Modus Ponens <Step 7> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 9> Implies(P, Implies(Q, P)) - Axiom 1 <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, P))) - Deducted <Step 11> Step 7 and Step 8 -> Q - Modus Ponens <Step 12> Step 9 and Step 10 -> Implies(Q, Implies(P, P)) - Modus Ponens <Step 13> Q - Deducted <Step 14> Implies(Q, Implies(P, P)) - Deducted <Step 15> Step 13 and Step 14 -> Implies(P, P) - Modus Ponens <Step 16> Implies(P, P) - Deducted

<Goal> Implies(P, P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, P))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q)) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Assumption <Step 5> P - Assumption <Step 6> Implies(P, Implies(P, Implies(Q, Implies(Q, P)))) - Assumption <Step 7> Step 1 and Step 2 -> Implies(Q, Q) - Modus Ponens <Step 8> Step 3 and Step 4 -> P - Modus Ponens <Step 9> Step 5 and Step 6 -> Implies(P, Implies(Q, Implies(Q, P))) - Modus Ponens <Step 10> Q - Assumption <Step 11> Implies(Q, Q) - Deducted <Step 12> P - Deducted <Step 13> Implies(P, Implies(Q, Implies(Q, P))) - Deducted <Step 14> Step 10 and Step 11 -> Q - Modus Ponens <Step 15> Step 12 and Step 13 -> Implies(Q, Implies(Q, P)) - Modus Ponens <Step 16> Q - Deducted <Step 17> Implies(Q, Implies(Q, P)) - Deducted <Step 18> Step 16 and Step 17 -> Implies(Q, P) - Modus Ponens <Step 19> Implies(Q, P) - Deducted

<Goal> Implies(Q, P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), P) <Assumption> P <Assumption> Implies(P, Implies(P, Implies(Q, Implies(Q, P)))) <Assumption> Q

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(P, P)) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Assumption <Step 5> P - Assumption <Step 6> Implies(P, Implies(P, Implies(P, Q))) - Assumption <Step 7> Step 1 and Step 2 -> Implies(P, P) - Modus Ponens <Step 8> Step 3 and Step 4 -> P - Modus Ponens <Step 9> Step 5 and Step 6 -> Implies(P, Implies(P, Q)) - Modus Ponens <Step 10> P - Assumption <Step 11> Implies(P, P) - Deducted <Step 12> P - Deducted <Step 13> Implies(P, Implies(P, Q)) - Deducted <Step 14> Step 10 and Step 11 -> P - Modus Ponens <Step 15> Step 12 and Step 13 -> Implies(P, Q) - Modus Ponens <Step 16> P - Deducted <Step 17> Implies(P, Q) - Deducted <Step 18> Step 16 and Step 17 -> Q - Modus Ponens <Step 19> Q - Deducted

<Goal> Q <Assumption> Q <Assumption> Implies(Q, Implies(P, P)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), P) <Assumption> P <Assumption> Implies(P, Implies(P, Implies(P, Q))) <Assumption> P

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Q, P)), P)) - Axiom 1 <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), P) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), P) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Axiom 1 <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), P) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Modus Ponens <Step 7> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, Q) - Modus Ponens <Step 10> Implies(P, Q) - Deducted

<Goal> Implies(P, Q) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(P, P) - Modus Ponens <Step 7> P - Assumption <Step 8> Implies(P, P) - Deducted <Step 9> Step 7 and Step 8 -> P - Modus Ponens <Step 10> P - Deducted

<Goal> P <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P))) <Assumption> P

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Q))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Q))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)) - Modus Ponens <Step 7> Implies(P, Implies(Q, P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Q, Q) - Modus Ponens <Step 10> Implies(Q, Q) - Deducted

<Goal> Implies(Q, Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Q))))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Q, P)), Q)) - Axiom 1 <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Q) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Q) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Modus Ponens <Step 10> Implies(P, Implies(Q, P)) - Axiom 1 <Step 11> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Deducted <Step 12> Step 10 and Step 11 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 13> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Q) - Assumption <Step 5> Q - Assumption <Step 6> Implies(Q, Implies(Q, Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))))) - Assumption <Step 7> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), P) - Modus Ponens <Step 8> Step 3 and Step 4 -> Q - Modus Ponens <Step 9> Step 5 and Step 6 -> Implies(Q, Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Modus Ponens <Step 10> Implies(P, Implies(Q, P)) - Axiom 1 <Step 11> Implies(Implies(P, Implies(Q, P)), P) - Deducted <Step 12> Q - Deducted <Step 13> Implies(Q, Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 14> Step 10 and Step 11 -> P - Modus Ponens <Step 15> Step 12 and Step 13 -> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 16> P - Deducted <Step 17> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted <Step 18> Step 16 and Step 17 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Modus Ponens <Step 19> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Deducted

<Goal> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) <Assumption> Q <Assumption> Implies(Q, Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, P))) - Assumption <Step 5> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Modus Ponens <Step 7> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 9> Implies(P, Implies(Q, P)) - Axiom 1 <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Deducted <Step 11> Step 7 and Step 8 -> Q - Modus Ponens <Step 12> Step 9 and Step 10 -> Implies(Q, P) - Modus Ponens <Step 13> Q - Deducted <Step 14> Implies(Q, P) - Deducted <Step 15> Step 13 and Step 14 -> P - Modus Ponens <Step 16> P - Deducted

<Goal> P <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, P)))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 7> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Modus Ponens <Step 10> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Deducted

<Goal> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))))

<Step 1> P - Assumption <Step 2> Implies(P, Q) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) - Assumption <Step 5> Q - Assumption <Step 6> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Assumption <Step 7> Step 1 and Step 2 -> Q - Modus Ponens <Step 8> Step 3 and Step 4 -> Implies(Q, P) - Modus Ponens <Step 9> Step 5 and Step 6 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Modus Ponens <Step 10> Q - Deducted <Step 11> Implies(Q, P) - Deducted <Step 12> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 13> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Deducted <Step 14> Step 10 and Step 11 -> P - Modus Ponens <Step 15> Step 12 and Step 13 -> Implies(P, Q) - Modus Ponens <Step 16> P - Deducted <Step 17> Implies(P, Q) - Deducted <Step 18> Step 16 and Step 17 -> Q - Modus Ponens <Step 19> Q - Deducted

<Goal> Q <Assumption> P <Assumption> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 7> Implies(P, Implies(Q, P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Q) - Assumption <Step 3> Implies(P, Implies(Q, P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(Q, P)) - Axiom 1 <Step 11> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted <Step 12> Step 10 and Step 11 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Modus Ponens <Step 13> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) <Assumption> Q <Assumption> Implies(Q, Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, P) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(Q, P)) - Axiom 1 <Step 11> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Deducted <Step 12> Step 10 and Step 11 -> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Modus Ponens <Step 13> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Deducted

<Goal> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) <Assumption> Q <Assumption> Implies(Q, P) <Assumption> Q <Assumption> Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(P, P) - Modus Ponens <Step 7> P - Assumption <Step 8> Implies(P, P) - Deducted <Step 9> Step 7 and Step 8 -> P - Modus Ponens <Step 10> P - Deducted

<Goal> P <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, P))) <Assumption> P

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(Q, P)) - Axiom 1 <Step 11> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Deducted <Step 12> Step 10 and Step 11 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Modus Ponens <Step 13> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Deducted

<Goal> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> P <Assumption> Implies(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q)) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Q) - Modus Ponens <Step 7> Implies(P, Implies(Q, P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Q, P)), Q) - Deducted <Step 9> Step 7 and Step 8 -> Q - Modus Ponens <Step 10> Q - Deducted

<Goal> Q <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q)))

<Step 1> Q - Assumption <Step 2> Implies(Q, P) - Assumption <Step 3> Implies(P, Implies(Q, P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) - Assumption <Step 5> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 6> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Assumption <Step 7> Implies(P, Implies(Q, P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))))) - Assumption <Step 9> Step 1 and Step 2 -> P - Modus Ponens <Step 10> Step 3 and Step 4 -> Implies(P, Q) - Modus Ponens <Step 11> Step 5 and Step 6 -> Q - Modus Ponens <Step 12> Step 7 and Step 8 -> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Modus Ponens <Step 13> P - Deducted <Step 14> Implies(P, Q) - Deducted <Step 15> Q - Deducted <Step 16> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Deducted <Step 17> Step 13 and Step 14 -> Q - Modus Ponens <Step 18> Step 15 and Step 16 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Modus Ponens <Step 19> Q - Deducted <Step 20> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Deducted <Step 21> Step 19 and Step 20 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Modus Ponens <Step 22> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) <Assumption> Q <Assumption> Implies(Q, P) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Modus Ponens <Step 7> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 8> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, P) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(P, Implies(P, Implies(P, Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Implies(P, Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(P, Implies(P, Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, Implies(P, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(P, Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(P, Implies(P, Implies(P, Implies(P, Q)))) <Assumption> Q <Assumption> Implies(Q, P) <Assumption> Q <Assumption> Implies(Q, Implies(P, Implies(P, Implies(P, Implies(P, Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted

<Goal> P <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Q, Implies(P, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 7> Implies(Q, Implies(P, Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Q, Implies(P, Implies(P, Implies(P, Q)))) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Q, Q)) - Axiom 1 <Step 3> Step 1 and Step 2 -> Implies(Q, Q) - Modus Ponens <Step 4> Q - Assumption <Step 5> Implies(Q, Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted

<Goal> Q <Assumption> Q <Assumption> Q

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q)))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), P) <Assumption> P <Assumption> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q)))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))))

<Step 1> P - Assumption <Step 2> Implies(P, Q) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, P))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(P, P)) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(P, P)) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, P) - Modus Ponens <Step 10> Implies(P, P) - Deducted

<Goal> Implies(P, P) <Assumption> P <Assumption> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, P)))

<Step 1> Q - Assumption <Step 2> Implies(Q, P) - Assumption <Step 3> Step 1 and Step 2 -> P - Modus Ponens <Step 4> P - Deducted <Step 5> Implies(P, Implies(Q, P)) - Axiom 1 <Step 6> Step 4 and Step 5 -> Implies(Q, P) - Modus Ponens <Step 7> Implies(Q, P) - Deducted

<Goal> Implies(Q, P) <Assumption> Q <Assumption> Implies(Q, P)

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Q) - Modus Ponens <Step 4> Q - Assumption <Step 5> Implies(Q, Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted

<Goal> Q <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q)) <Assumption> Q

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P)) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), P) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Q, P)), P) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), P) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P)))

<Step 1> Q - Assumption <Step 2> Implies(Q, Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), P) <Assumption> Q <Assumption> Implies(Q, Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(P, Implies(Q, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 7> Implies(P, Implies(Q, Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(P, Implies(Q, Implies(P, Implies(P, Q)))) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Q, Implies(P, Implies(P, Q))))))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Q, P)), P)) - Axiom 1 <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), P) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted

<Goal> P <Assumption> P

<Step 1> Q - Assumption <Step 2> Implies(Q, Q) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(Q, Implies(Q, Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Q, Implies(Q, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(Q, Implies(Q, Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Q, Implies(Q, Implies(P, Implies(P, Q)))) <Assumption> Q <Assumption> Implies(Q, Q) <Assumption> P <Assumption> Implies(P, Implies(Q, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted

<Goal> Q <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q)))) <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q)))) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q)))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), P) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(P, Implies(P, Q))) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Implies(P, Q)) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(P, Q)) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, Q) - Modus Ponens <Step 10> Implies(P, Q) - Deducted

<Goal> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Q, P)), P) <Assumption> Q <Assumption> Implies(Q, Implies(P, Implies(P, Q)))

<Step 1> Q - Assumption <Step 2> Implies(Q, Q) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, P) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, P) - Deducted <Step 9> Step 7 and Step 8 -> P - Modus Ponens <Step 10> P - Deducted

<Goal> P <Assumption> Q <Assumption> Implies(Q, Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P))

<Step 1> P - Assumption <Step 2> Implies(P, Q) - Assumption <Step 3> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted <Step 5> Implies(Q, Implies(Q, Q)) - Axiom 1 <Step 6> Step 4 and Step 5 -> Implies(Q, Q) - Modus Ponens <Step 7> Implies(Q, Q) - Deducted

<Goal> Implies(Q, Q) <Assumption> P <Assumption> Implies(P, Q)

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Assumption <Step 3> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted <Step 5> Implies(Q, Implies(Implies(P, Implies(Q, P)), Q)) - Axiom 1 <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Q) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Q, P)), Q) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q)

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> Implies(P, Implies(Q, P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Q, P)), P)) - Axiom 1 <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), P) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted

<Goal> P <Assumption> P

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 7> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), P) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(P, Q)) - Axiom 1 <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Q) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Q) - Deducted <Step 9> Step 7 and Step 8 -> Q - Modus Ponens <Step 10> Q - Deducted

<Goal> Q <Assumption> Implies(Implies(P, Implies(Q, P)), P) <Assumption> Q

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))))

<Step 1> Q - Assumption <Step 2> Implies(Q, P) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q)))) <Assumption> Q <Assumption> Implies(Q, P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Q))))))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 7> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, P)) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, P) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, P) - Deducted <Step 9> Step 7 and Step 8 -> P - Modus Ponens <Step 10> P - Deducted

<Goal> P <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, P))

<Step 1> P - Assumption <Step 2> Implies(P, Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) <Assumption> P <Assumption> Implies(P, Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted

<Goal> Q <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q))

<Step 1> Q - Assumption <Step 2> Implies(Q, P) - Assumption <Step 3> Implies(P, Implies(Q, P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) <Assumption> Q <Assumption> Implies(Q, P) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q)))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Q))))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, P) - Modus Ponens <Step 4> P - Assumption <Step 5> Implies(P, P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted

<Goal> P <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) <Assumption> P

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, P)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, P))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, P))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Q, Implies(P, P)) - Modus Ponens <Step 7> Implies(Q, Implies(P, P)) - Deducted

<Goal> Implies(Q, Implies(P, P)) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, P))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Q) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted

<Goal> Q <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, P)))) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, P))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, P))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, P))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, P) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, P) - Deducted <Step 9> Step 7 and Step 8 -> P - Modus Ponens <Step 10> P - Deducted

<Goal> P <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> P

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, P)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, P))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, P))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Q, Implies(Q, P)) - Modus Ponens <Step 7> Implies(Q, Implies(Q, P)) - Deducted

<Goal> Implies(Q, Implies(Q, P)) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, P))))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted

<Goal> Q <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, P)))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, P))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, P))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> P <Assumption> Implies(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, P))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> P <Assumption> Implies(P, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(P, Q) - Modus Ponens <Step 7> Implies(P, Q) - Deducted

<Goal> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted

<Goal> P <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P))

<Step 1> P - Assumption <Step 2> Implies(P, P) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Q, Q))) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Implies(Q, Q)) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Q, Q)) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Q, Q) - Modus Ponens <Step 10> Implies(Q, Q) - Deducted

<Goal> Implies(Q, Q) <Assumption> P <Assumption> Implies(P, P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Q, Q)))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Q) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Q, P)), Q) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)))

<Step 1> P - Assumption <Step 2> Implies(P, Q) - Assumption <Step 3> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted <Step 5> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Axiom 1 <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> P <Assumption> Implies(P, Q)

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), P) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted

<Goal> P <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), P)))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), P))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), P))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) - Modus Ponens <Step 10> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) - Deducted

<Goal> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), P))))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted

<Goal> Q <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P)) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P)) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P)) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), P))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), P)))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), P))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), P))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), P)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), P)) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), P)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), P))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, P) - Modus Ponens <Step 4> P - Assumption <Step 5> Implies(P, P) - Deducted <Step 6> Step 4 and Step 5 -> P - Modus Ponens <Step 7> P - Deducted

<Goal> P <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)) <Assumption> P

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Modus Ponens <Step 7> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Deducted

<Goal> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P))))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted

<Goal> Q <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> Implies(P, Implies(Q, P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q)))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(P, Implies(P, Q))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(P, Implies(P, Q))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, Implies(P, Q)) - Modus Ponens <Step 10> Implies(P, Implies(P, Q)) - Deducted

<Goal> Implies(P, Implies(P, Q)) <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, P)) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, P) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, P) - Deducted <Step 9> Step 7 and Step 8 -> P - Modus Ponens <Step 10> P - Deducted

<Goal> P <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, P))

<Step 1> Q - Assumption <Step 2> Implies(Q, P) - Assumption <Step 3> Implies(P, Implies(Q, P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) - Assumption <Step 5> Step 1 and Step 2 -> P - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(P, Q) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Q) - Deducted <Step 9> Step 7 and Step 8 -> Q - Modus Ponens <Step 10> Q - Deducted

<Goal> Q <Assumption> Q <Assumption> Implies(Q, P) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(P, Q))

<Step 1> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) - Deducted

<Goal> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))

<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Q) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Deducted

<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> P <Assumption> Implies(P, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q))))

<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(P, P) - Modus Ponens <Step 7> Implies(P, P) - Deducted

<Goal> Implies(P, P) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, P)))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Q, P) - Modus Ponens <Step 7> Implies(Q, P) - Deducted

<Goal> Implies(Q, P) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, P)))

<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, P), P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted <Step 6> Step 4 and Step 5 -> Q - Modus Ponens <Step 7> Q - Deducted

<Goal> Q <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q))