Datasets:
yuan-xia
/
proofs-data-trial

Dataset card Viewer Files Community
1
input
stringclasses
1 value
instruction
stringlengths
23
488
output
stringlengths
128
2.43k
<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q))))) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q))))))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), 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(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(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(P, Q) <Assumption> 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(P, Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Q) - Modus Ponens <Step 4> Implies(P, Q) - Deducted
<Goal> P <Assumption> 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> Step 1 and Step 2 -> P - Modus Ponens <Step 4> P - Deducted
<Goal> Implies(Q, Q) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, 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> Implies(Q, Q) - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), 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(Implies(P, Implies(Q, 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)), 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)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted
<Goal> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) <Assumption> P <Assumption> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Deducted
<Goal> P <Assumption> Q <Assumption> Implies(Q, 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
<Goal> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))))
<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Q <Assumption> 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> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(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, 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(Q, 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(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(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, Implies(P, 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(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(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(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))) <Assumption> Q <Assumption> Implies(Q, Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))))
<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(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(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, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))) - Deducted
<Goal> P <Assumption> Q <Assumption> Implies(Q, 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
<Goal> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))))
<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Q <Assumption> 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> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), 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(Q, P)), Implies(Implies(P, Implies(Implies(P, P), 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(Q, 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(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, P)), 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(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> 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(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(P, Implies(P, Q))))) - Deducted
<Goal> P <Assumption> 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> Step 1 and Step 2 -> P - Modus Ponens <Step 4> P - Deducted
<Goal> Implies(Q, Implies(P, P)) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, P)))
<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(P, P)) - Modus Ponens <Step 4> Implies(Q, Implies(P, P)) - Deducted
<Goal> 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
<Goal> Implies(Implies(P, Implies(Q, P)), Implies(P, P)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), 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)), 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(Implies(P, Implies(Q, P)), Implies(P, P)) - Deducted
<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)))
<Step 1> P - Assumption <Step 2> Implies(P, 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(Implies(P, Implies(Implies(P, P), P)), Implies(P, P)) - Deducted
<Goal> 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
<Goal> Implies(Q, Implies(Q, P)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, 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, Implies(Q, P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(Q, P)) - Modus Ponens <Step 4> Implies(Q, Implies(Q, P)) - Deducted
<Goal> Q <Assumption> 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> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, 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, P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Deducted
<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, P)) - Deducted
<Goal> Implies(P, Q) <Assumption> 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(P, Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Q) - Modus Ponens <Step 4> Implies(P, Q) - Deducted
<Goal> P <Assumption> Q <Assumption> Implies(Q, 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
<Goal> Implies(Q, Q) <Assumption> Implies(Implies(P, Implies(Implies(P, P), 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(Q, Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Q) - Modus Ponens <Step 4> Implies(Q, Q) - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), 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(Implies(P, Implies(Q, 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)), 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)), Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted
<Goal> 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
<Goal> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, 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(Q, Implies(Implies(P, Implies(Q, P)), P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) - Modus Ponens <Step 4> Implies(Q, Implies(Implies(P, Implies(Q, P)), P)) - Deducted
<Goal> 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
<Goal> Implies(Implies(P, Implies(Q, P)), 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> 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(Implies(P, Implies(Q, 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(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(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), P)) - Deducted
<Goal> 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
<Goal> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) <Assumption> P <Assumption> Implies(P, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Modus Ponens <Step 4> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Deducted
<Goal> Q <Assumption> 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> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) <Assumption> Implies(Implies(P, Implies(Q, 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)), 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(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, 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> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), 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(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(Implies(P, P), P)), P)) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Deducted
<Goal> Implies(P, Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> P <Assumption> Implies(P, Implies(P, Implies(P, Implies(Q, Implies(P, Implies(P, Q))))))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(P, Implies(P, Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> 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
<Goal> Implies(Q, Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(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)), Implies(Q, Implies(P, Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Q, Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> 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
<Goal> Implies(Implies(P, Implies(Q, P)), 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)), 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(Implies(P, Implies(Q, P)), Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Q, Implies(P, Implies(P, Q))))))
<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(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(P, Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(P, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(P, P) <Assumption> Q <Assumption> Implies(Q, Implies(P, P))
<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(P, P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, P) - Modus Ponens <Step 4> Implies(P, P) - Deducted
<Goal> Implies(Q, P) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Q, P))
<Step 1> Implies(P, Implies(Q, P)) - Axiom 1 <Step 2> Implies(Implies(P, Implies(Q, P)), Implies(Q, P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, P) - Modus Ponens <Step 4> Implies(Q, P) - Deducted
<Goal> 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
<Goal> Implies(Implies(P, Implies(Q, P)), P) <Assumption> P
<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(Implies(P, Implies(Q, P)), P) - Deducted
<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), P) <Assumption> P
<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), P)) - Axiom 1 <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), P) - Deducted
<Goal> Implies(P, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> P <Assumption> Implies(P, Implies(P, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(P, Implies(Q, Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> P <Assumption> P <Assumption> Implies(P, P)
<Step 1> P - Assumption <Step 2> Implies(P, P) - Assumption <Step 3> Step 1 and Step 2 -> P - Modus Ponens <Step 4> P - Deducted
<Goal> Implies(Q, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> Q <Assumption> Implies(Q, Implies(Q, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))))
<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Q, Implies(Q, Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Q, Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Q <Assumption> 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> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(P, Q) <Assumption> 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(P, Q)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Q) - Modus Ponens <Step 4> Implies(P, Q) - Deducted
<Goal> 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
<Goal> Implies(Q, Q) <Assumption> 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> Implies(Q, Q) - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), Q) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Q))
<Step 1> P - Assumption <Step 2> Implies(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(Implies(P, Implies(Q, P)), Q) - Deducted
<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Q) <Assumption> Q
<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Axiom 1 <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Q) - Deducted
<Goal> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(P, Implies(Implies(P, Implies(Q, 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)), Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> P <Assumption> P <Assumption> Implies(P, P)
<Step 1> P - Assumption <Step 2> Implies(P, P) - Assumption <Step 3> Step 1 and Step 2 -> P - Modus Ponens <Step 4> P - Deducted
<Goal> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), 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(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Q, Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Q <Assumption> 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> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(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(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(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(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(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, 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)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, 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(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) <Assumption> P <Assumption> Implies(P, Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(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(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, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> P <Assumption> Q <Assumption> Implies(Q, 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
<Goal> Implies(Q, 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(Q, Implies(Implies(P, Implies(Implies(P, P), 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(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q)))))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Q <Assumption> 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> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), 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(Q, P)), Implies(Implies(P, Implies(Implies(P, P), 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(Q, 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(Q, P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, P)), 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(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(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(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Implies(P, Implies(P, Q))))) - Deducted
<Goal> Implies(P, Implies(P, Q)) <Assumption> P <Assumption> Implies(P, Implies(P, Implies(P, Q)))
<Step 1> P - Assumption <Step 2> Implies(P, Implies(P, Implies(P, Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Implies(P, Q)) - Modus Ponens <Step 4> Implies(P, Implies(P, Q)) - Deducted
<Goal> P <Assumption> P <Assumption> Implies(P, P)
<Step 1> P - Assumption <Step 2> Implies(P, P) - Assumption <Step 3> Step 1 and Step 2 -> P - Modus Ponens <Step 4> P - Deducted
<Goal> Q <Assumption> 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> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> 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(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, Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(P, Q)) - Deducted
<Goal> 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(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(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(Implies(P, Implies(Implies(P, P), P)), Implies(P, Q)) - Deducted
<Goal> Implies(P, P) <Assumption> 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(P, P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, P) - Modus Ponens <Step 4> Implies(P, P) - Deducted
<Goal> Implies(Q, P) <Assumption> 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(Q, P)) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Q, P) - Modus Ponens <Step 4> Implies(Q, P) - Deducted
<Goal> 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
<Goal> Implies(Implies(P, Implies(Q, P)), P) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Q, P)), P))
<Step 1> Q - Assumption <Step 2> Implies(Q, 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(Implies(P, Implies(Q, 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> 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(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(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(P, Implies(Q, Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Implies(Q, Q)) - Modus Ponens <Step 4> Implies(P, Implies(Q, Q)) - Deducted
<Goal> P <Assumption> Q <Assumption> Implies(Q, 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
<Goal> 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
<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)) <Assumption> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, 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(Q, P)), Implies(Q, Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)) - Modus Ponens <Step 4> Implies(Implies(P, Implies(Q, P)), Implies(Q, Q)) - Deducted
<Goal> Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q)) <Assumption> Q <Assumption> Implies(Q, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Q, Q)))
<Step 1> Q - Assumption <Step 2> Implies(Q, 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(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, Implies(P, Implies(Implies(P, Implies(Q, P)), Q)))
<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(P, Implies(Implies(P, Implies(Q, P)), Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Implies(Implies(P, Implies(Q, P)), Q)) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, Implies(Q, P)), Q)) - Deducted
<Goal> P <Assumption> Q <Assumption> Implies(Q, 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
<Goal> Q <Assumption> 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> Step 1 and Step 2 -> Q - Modus Ponens <Step 4> Q - Deducted
<Goal> Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)) <Assumption> P <Assumption> Implies(P, Implies(Implies(P, Implies(Q, P)), Implies(Implies(P, Implies(Q, P)), Q)))
<Step 1> P - Assumption <Step 2> Implies(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(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, Implies(Implies(P, Implies(Implies(P, P), P)), Implies(Implies(P, Implies(Q, P)), Q)))
<Step 1> P - Assumption <Step 2> Implies(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(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> Q <Assumption> Implies(Q, Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)))
<Step 1> Q - Assumption <Step 2> Implies(Q, Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q))) - Assumption <Step 3> Step 1 and Step 2 -> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Modus Ponens <Step 4> Implies(P, Implies(Implies(P, Implies(Implies(P, P), P)), Q)) - Deducted
<Goal> P <Assumption> Q <Assumption> Implies(Q, 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