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yuan-xia
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proofs-data-trial

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train · 10.2k rows

input stringclasses 1 value	instruction stringlengths 23 488	output stringlengths 128 2.43k
	<Goal> Implies(Implies(P, Implies(Q, P)), P) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), 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)), Implies(Implies(P, Implies(Implies(P, P), 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)), 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)), 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(Implies(P, P), P)), P) <Assumption> P <Assumption> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)))	<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(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(P, Implies(Q, Q)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Q, 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(P, Implies(Q, Q)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(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(P, Implies(Q, Q))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(P, Implies(Q, Q)) - Modus Ponens <Step 7> Implies(P, Implies(Q, Q)) - Deducted
	<Goal> P <Assumption> Implies(Implies(P, Implies(Implies(P, P), 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)), 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> Q <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)) <Assumption> Q	<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, 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> Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)) <Assumption> Q <Assumption> Implies(Q, P) <Assumption> P <Assumption> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Q))))	<Step 1> Q - Assumption <Step 2> Implies(Q, P) - Assumption <Step 3> P - Assumption <Step 4> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, 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(Implies(P, Implies(Q, P)), Implies(Q, Q))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Q))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)) - Deducted
	<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q)) <Assumption> P <Assumption> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, 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(Implies(P, P), P)), Implies(Q, 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(Q, Q))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q)) - Deducted
	<Goal> Implies(P, Implies(Implies(P, Implies(Q, P)), Q)) <Assumption> Q <Assumption> Implies(Q, Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Q))))	<Step 1> Q - Assumption <Step 2> Implies(Q, Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, 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)), Q))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Q))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Q)) - Modus Ponens <Step 10> Implies(P, Implies(Implies(P, Implies(Q, P)), Q)) - Deducted
	<Goal> P <Assumption> Q <Assumption> Implies(Q, Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, P))	<Step 1> Q - Assumption <Step 2> Implies(Q, 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> Q <Assumption> P <Assumption> Implies(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)), 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> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)) <Assumption> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, 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(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, 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(Q, P)), Q))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)) - Deducted
	<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q)) <Assumption> P <Assumption> Implies(P, Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q))))	<Step 1> P - Assumption <Step 2> Implies(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)), 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)), 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)), Q))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q)) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q)) - Deducted
	<Goal> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), 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)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q))) - Modus Ponens <Step 4> Implies(P, Implies(Q, P)) - Axiom 1 <Step 5> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Modus Ponens <Step 7> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Deducted
	<Goal> P <Assumption> Q <Assumption> Implies(Q, Q) <Assumption> P	<Step 1> Q - Assumption <Step 2> Implies(Q, 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> Q <Assumption> Q	<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Q, P)), Q)) - Axiom 1 <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> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) <Assumption> P <Assumption> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), 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(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(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Deducted
	<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), 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(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)), Implies(Implies(P, Implies(Implies(P, P), 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)), 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)), 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)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Deducted
	<Goal> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, 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)), 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(P, Implies(P, 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, Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted
	<Goal> P <Assumption> P <Assumption> P	<Step 1> P - Assumption <Step 2> Implies(P, Implies(P, P)) - Axiom 1 <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> Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Q, Implies(P, 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)), Q) - Assumption <Step 3> Q - Assumption <Step 4> Implies(Q, Implies(Q, Implies(Q, Implies(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(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 10> Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, 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)), 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> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))))	<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(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, 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, 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, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted
	<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, 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(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(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, 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(Implies(P, Implies(Q, 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(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(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted
	<Goal> Implies(P, P) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> 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)), 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(Q, P) <Assumption> P <Assumption> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Q, P)))	<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(Q, P))) - Assumption <Step 5> Step 1 and Step 2 -> Q - Modus Ponens <Step 6> Step 3 and Step 4 -> Implies(Q, Implies(Q, P)) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(Q, P)) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Q, P) - Modus Ponens <Step 10> Implies(Q, P) - Deducted
	<Goal> Q <Assumption> P <Assumption> Implies(P, P) <Assumption> Q	<Step 1> P - Assumption <Step 2> Implies(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> Implies(Implies(P, Implies(Q, P)), P) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), 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)), Implies(Implies(P, Implies(Implies(P, P), 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)), 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)), 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(Implies(P, P), P)), P) <Assumption> 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)), P) - Assumption <Step 3> Step 1 and Step 2 -> P - Modus Ponens <Step 4> P - Deducted <Step 5> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Axiom 1 <Step 6> Step 4 and Step 5 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 7> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted
	<Goal> Implies(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) <Assumption> P <Assumption> Implies(P, Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(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(Q, P)) - Axiom 1 <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(Q, 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(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Modus Ponens <Step 7> Q - Deducted <Step 8> Implies(Q, Implies(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted
	<Goal> P <Assumption> P <Assumption> P	<Step 1> P - Assumption <Step 2> Implies(P, Implies(P, P)) - Axiom 1 <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> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))))	<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, 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(Q, 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(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 7> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted
	<Goal> Q <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, 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)), 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> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) <Assumption> P <Assumption> Implies(P, Q) <Assumption> P <Assumption> Implies(P, Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))))	<Step 1> P - Assumption <Step 2> Implies(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(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(Q, 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(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Deducted <Step 9> Step 7 and Step 8 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 10> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted
	<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Q, 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(Q, 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)), Implies(Implies(P, Implies(Implies(P, P), 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(Implies(P, P), 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(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(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(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted
	<Goal> Implies(P, Q) <Assumption> P <Assumption> Implies(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(P, Q)) - Axiom 1 <Step 6> Step 4 and Step 5 -> Implies(P, Q) - Modus Ponens <Step 7> Implies(P, Q) - Deducted
	<Goal> P <Assumption> 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)), 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> Implies(Q, Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), 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(Implies(P, P), P)), Implies(Q, Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), 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(Q, Q)) - Deducted <Step 6> Step 4 and Step 5 -> Implies(Q, Q) - Modus Ponens <Step 7> Implies(Q, Q) - Deducted
	<Goal> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q)))	<Step 1> Q - Assumption <Step 2> Implies(Q, 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(Implies(P, Implies(Q, P)), Q) - Deducted
	<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Q <Assumption> Implies(Q, Q)	<Step 1> Q - Assumption <Step 2> Implies(Q, 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(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) <Assumption> Q <Assumption> Implies(Q, P) <Assumption> Q <Assumption> Implies(Q, Implies(P, Implies(P, Implies(Implies(P, Implies(Q, 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(P, Implies(Implies(P, Implies(Q, P)), 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, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q))))) - Modus Ponens <Step 7> P - Deducted <Step 8> Implies(P, Implies(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(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Modus Ponens <Step 10> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Q)))) - Deducted