%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LCL109-4 : TPTP v9.3.1. Released v1.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n001.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 11:49:03 AM UTC 2026
% Result : Unsatisfiable 275.05s 40.09s
% Output : Proof 276.49s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.02 % Problem : LCL109-4 : TPTP v9.3.1. Released v1.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.09/0.37 % Computer : n001.cluster.edu
% 0.09/0.37 % Model : x86_64 x86_64
% 0.09/0.37 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.37 % Memory : 8046.5625MB
% 0.09/0.37 % OS : Linux 6.8.0-71-generic
% 0.09/0.37 % CPULimit : 300
% 0.09/0.37 % WCLimit : 300
% 0.09/0.37 % DateTime : Sun Sep 27 15:28:00 UTC 2026
% 0.09/0.37 % CPUTime :
% 0.09/0.37 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 275.05/40.09 Command-line arguments: --no-flatten-goal
% 275.05/40.09
% 275.05/40.09 % SZS status Unsatisfiable
% 275.05/40.09
% 275.77/40.19 % SZS output start Proof
% 275.77/40.19 Axiom 1 (wajsberg_1): implies(truth, X) = X.
% 275.77/40.19 Axiom 2 (big_hat_definition): big_hat(X, Y) = not(big_V(not(X), not(Y))).
% 275.77/40.19 Axiom 3 (big_V_definition): big_V(X, Y) = implies(implies(X, Y), Y).
% 275.77/40.19 Axiom 4 (wajsberg_3): implies(implies(X, Y), Y) = implies(implies(Y, X), X).
% 275.77/40.19 Axiom 5 (wajsberg_4): implies(implies(not(X), not(Y)), implies(Y, X)) = truth.
% 275.77/40.19 Axiom 6 (wajsberg_2): implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))) = truth.
% 275.77/40.19
% 275.77/40.19 Lemma 7: big_V(Y, X) = big_V(X, Y).
% 275.77/40.19 Proof:
% 275.77/40.19 big_V(Y, X)
% 275.77/40.19 = { by axiom 3 (big_V_definition) }
% 275.77/40.19 implies(implies(Y, X), X)
% 275.77/40.19 = { by axiom 4 (wajsberg_3) R->L }
% 275.77/40.19 implies(implies(X, Y), Y)
% 275.77/40.19 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.19 big_V(X, Y)
% 275.77/40.19
% 275.77/40.19 Lemma 8: implies(X, big_V(X, Y)) = truth.
% 275.77/40.19 Proof:
% 275.77/40.19 implies(X, big_V(X, Y))
% 275.77/40.19 = { by axiom 3 (big_V_definition) }
% 275.77/40.19 implies(X, implies(implies(X, Y), Y))
% 275.77/40.19 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.19 implies(X, implies(implies(X, Y), implies(truth, Y)))
% 275.77/40.19 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.19 implies(implies(truth, X), implies(implies(X, Y), implies(truth, Y)))
% 275.77/40.19 = { by axiom 6 (wajsberg_2) }
% 275.77/40.19 truth
% 275.77/40.19
% 275.77/40.19 Lemma 9: big_V(X, truth) = truth.
% 275.77/40.19 Proof:
% 275.77/40.19 big_V(X, truth)
% 275.77/40.19 = { by lemma 7 }
% 275.77/40.19 big_V(truth, X)
% 275.77/40.19 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.19 implies(truth, big_V(truth, X))
% 275.77/40.19 = { by lemma 8 }
% 275.77/40.19 truth
% 275.77/40.19
% 275.77/40.19 Lemma 10: implies(X, truth) = truth.
% 275.77/40.19 Proof:
% 275.77/40.19 implies(X, truth)
% 275.77/40.19 = { by lemma 9 R->L }
% 275.77/40.19 implies(X, big_V(X, truth))
% 275.77/40.20 = { by lemma 8 }
% 275.77/40.20 truth
% 275.77/40.20
% 275.77/40.20 Lemma 11: big_V(X, implies(not(X), not(truth))) = X.
% 275.77/40.20 Proof:
% 275.77/40.20 big_V(X, implies(not(X), not(truth)))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 big_V(implies(not(X), not(truth)), X)
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(implies(not(X), not(truth)), X), X)
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(implies(implies(not(X), not(truth)), implies(truth, X)), X)
% 275.77/40.20 = { by axiom 5 (wajsberg_4) }
% 275.77/40.20 implies(truth, X)
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 X
% 275.77/40.20
% 275.77/40.20 Lemma 12: big_V(implies(X, Y), implies(not(Y), not(X))) = implies(X, Y).
% 275.77/40.20 Proof:
% 275.77/40.20 big_V(implies(X, Y), implies(not(Y), not(X)))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 big_V(implies(not(Y), not(X)), implies(X, Y))
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(implies(not(Y), not(X)), implies(X, Y)), implies(X, Y))
% 275.77/40.20 = { by axiom 5 (wajsberg_4) }
% 275.77/40.20 implies(truth, implies(X, Y))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 implies(X, Y)
% 275.77/40.20
% 275.77/40.20 Lemma 13: implies(X, implies(Y, X)) = truth.
% 275.77/40.20 Proof:
% 275.77/40.20 implies(X, implies(Y, X))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(implies(truth, X), implies(Y, X))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(truth, implies(implies(truth, X), implies(Y, X)))
% 275.77/40.20 = { by lemma 10 R->L }
% 275.77/40.20 implies(implies(Y, truth), implies(implies(truth, X), implies(Y, X)))
% 275.77/40.20 = { by axiom 6 (wajsberg_2) }
% 275.77/40.20 truth
% 275.77/40.20
% 275.77/40.20 Lemma 14: implies(implies(X, Y), implies(X, big_V(Y, Z))) = truth.
% 275.77/40.20 Proof:
% 275.77/40.20 implies(implies(X, Y), implies(X, big_V(Y, Z)))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(implies(X, Y), implies(truth, implies(X, big_V(Y, Z))))
% 275.77/40.20 = { by lemma 8 R->L }
% 275.77/40.20 implies(implies(X, Y), implies(implies(Y, big_V(Y, Z)), implies(X, big_V(Y, Z))))
% 275.77/40.20 = { by axiom 6 (wajsberg_2) }
% 275.77/40.20 truth
% 275.77/40.20
% 275.77/40.20 Lemma 15: implies(X, big_V(Y, implies(Z, X))) = truth.
% 275.77/40.20 Proof:
% 275.77/40.20 implies(X, big_V(Y, implies(Z, X)))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 implies(X, big_V(implies(Z, X), Y))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(truth, implies(X, big_V(implies(Z, X), Y)))
% 275.77/40.20 = { by lemma 13 R->L }
% 275.77/40.20 implies(implies(X, implies(Z, X)), implies(X, big_V(implies(Z, X), Y)))
% 275.77/40.20 = { by lemma 14 }
% 275.77/40.20 truth
% 275.77/40.20
% 275.77/40.20 Lemma 16: implies(not(X), implies(X, Y)) = truth.
% 275.77/40.20 Proof:
% 275.77/40.20 implies(not(X), implies(X, Y))
% 275.77/40.20 = { by lemma 12 R->L }
% 275.77/40.20 implies(not(X), big_V(implies(X, Y), implies(not(Y), not(X))))
% 275.77/40.20 = { by lemma 15 }
% 275.77/40.20 truth
% 275.77/40.20
% 275.77/40.20 Lemma 17: implies(not(not(X)), X) = truth.
% 275.77/40.20 Proof:
% 275.77/40.20 implies(not(not(X)), X)
% 275.77/40.20 = { by lemma 11 R->L }
% 275.77/40.20 implies(not(not(X)), big_V(X, implies(not(X), not(truth))))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 implies(not(not(X)), big_V(implies(not(X), not(truth)), X))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(truth, implies(not(not(X)), big_V(implies(not(X), not(truth)), X)))
% 275.77/40.20 = { by lemma 16 R->L }
% 275.77/40.20 implies(implies(not(not(X)), implies(not(X), not(truth))), implies(not(not(X)), big_V(implies(not(X), not(truth)), X)))
% 275.77/40.20 = { by lemma 14 }
% 275.77/40.20 truth
% 275.77/40.20
% 275.77/40.20 Lemma 18: not(not(X)) = X.
% 275.77/40.20 Proof:
% 275.77/40.20 not(not(X))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(truth, not(not(X)))
% 275.77/40.20 = { by axiom 5 (wajsberg_4) R->L }
% 275.77/40.20 implies(implies(implies(not(not(not(X))), not(X)), implies(X, not(not(X)))), not(not(X)))
% 275.77/40.20 = { by lemma 17 }
% 275.77/40.20 implies(implies(truth, implies(X, not(not(X)))), not(not(X)))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 implies(implies(X, not(not(X))), not(not(X)))
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 big_V(X, not(not(X)))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 big_V(not(not(X)), X)
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(not(not(X)), X), X)
% 275.77/40.20 = { by lemma 17 }
% 275.77/40.20 implies(truth, X)
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 X
% 275.77/40.20
% 275.77/40.20 Lemma 19: big_V(X, X) = X.
% 275.77/40.20 Proof:
% 275.77/40.20 big_V(X, X)
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 big_V(X, implies(truth, X))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 big_V(implies(truth, X), X)
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(implies(truth, X), X), X)
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 implies(big_V(truth, X), X)
% 275.77/40.20 = { by lemma 7 R->L }
% 275.77/40.20 implies(big_V(X, truth), X)
% 275.77/40.20 = { by lemma 9 }
% 275.77/40.20 implies(truth, X)
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 X
% 275.77/40.20
% 275.77/40.20 Lemma 20: big_hat(X, X) = not(not(X)).
% 275.77/40.20 Proof:
% 275.77/40.20 big_hat(X, X)
% 275.77/40.20 = { by axiom 2 (big_hat_definition) }
% 275.77/40.20 not(big_V(not(X), not(X)))
% 275.77/40.20 = { by lemma 19 }
% 275.77/40.20 not(not(X))
% 275.77/40.20
% 275.77/40.20 Lemma 21: big_hat(Y, X) = big_hat(X, Y).
% 275.77/40.20 Proof:
% 275.77/40.20 big_hat(Y, X)
% 275.77/40.20 = { by axiom 2 (big_hat_definition) }
% 275.77/40.20 not(big_V(not(Y), not(X)))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 not(big_V(not(X), not(Y)))
% 275.77/40.20 = { by axiom 2 (big_hat_definition) R->L }
% 275.77/40.20 big_hat(X, Y)
% 275.77/40.20
% 275.77/40.20 Lemma 22: implies(X, not(truth)) = not(X).
% 275.77/40.20 Proof:
% 275.77/40.20 implies(X, not(truth))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(truth, implies(X, not(truth)))
% 275.77/40.20 = { by lemma 16 R->L }
% 275.77/40.20 implies(implies(not(X), implies(X, not(truth))), implies(X, not(truth)))
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 big_V(not(X), implies(X, not(truth)))
% 275.77/40.20 = { by lemma 18 R->L }
% 275.77/40.20 big_V(not(X), implies(not(not(X)), not(truth)))
% 275.77/40.20 = { by lemma 11 }
% 275.77/40.20 not(X)
% 275.77/40.20
% 275.77/40.20 Lemma 23: implies(not(X), not(Y)) = implies(Y, X).
% 275.77/40.20 Proof:
% 275.77/40.20 implies(not(X), not(Y))
% 275.77/40.20 = { by lemma 22 R->L }
% 275.77/40.20 implies(not(X), implies(Y, not(truth)))
% 275.77/40.20 = { by lemma 22 R->L }
% 275.77/40.20 implies(implies(X, not(truth)), implies(Y, not(truth)))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(truth, implies(implies(X, not(truth)), implies(Y, not(truth))))
% 275.77/40.20 = { by axiom 6 (wajsberg_2) R->L }
% 275.77/40.20 implies(implies(implies(Y, X), implies(implies(X, not(truth)), implies(Y, not(truth)))), implies(implies(X, not(truth)), implies(Y, not(truth))))
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 big_V(implies(Y, X), implies(implies(X, not(truth)), implies(Y, not(truth))))
% 275.77/40.20 = { by lemma 22 }
% 275.77/40.20 big_V(implies(Y, X), implies(not(X), implies(Y, not(truth))))
% 275.77/40.20 = { by lemma 22 }
% 275.77/40.20 big_V(implies(Y, X), implies(not(X), not(Y)))
% 275.77/40.20 = { by lemma 12 }
% 275.77/40.20 implies(Y, X)
% 275.77/40.20
% 275.77/40.20 Lemma 24: implies(X, not(Y)) = implies(Y, not(X)).
% 275.77/40.20 Proof:
% 275.77/40.20 implies(X, not(Y))
% 275.77/40.20 = { by lemma 23 R->L }
% 275.77/40.20 implies(not(not(Y)), not(X))
% 275.77/40.20 = { by lemma 18 }
% 275.77/40.20 implies(Y, not(X))
% 275.77/40.20
% 275.77/40.20 Lemma 25: implies(not(X), Y) = implies(not(Y), X).
% 275.77/40.20 Proof:
% 275.77/40.20 implies(not(X), Y)
% 275.77/40.20 = { by lemma 23 R->L }
% 275.77/40.20 implies(not(Y), not(not(X)))
% 275.77/40.20 = { by lemma 18 }
% 275.77/40.20 implies(not(Y), X)
% 275.77/40.20
% 275.77/40.20 Lemma 26: not(big_V(X, Y)) = big_hat(not(X), not(Y)).
% 275.77/40.20 Proof:
% 275.77/40.20 not(big_V(X, Y))
% 275.77/40.20 = { by lemma 18 R->L }
% 275.77/40.20 not(big_V(X, not(not(Y))))
% 275.77/40.20 = { by lemma 18 R->L }
% 275.77/40.20 not(big_V(not(not(X)), not(not(Y))))
% 275.77/40.20 = { by axiom 2 (big_hat_definition) R->L }
% 275.77/40.20 big_hat(not(X), not(Y))
% 275.77/40.20
% 275.77/40.20 Lemma 27: not(big_hat(X, Y)) = big_V(not(X), not(Y)).
% 275.77/40.20 Proof:
% 275.77/40.20 not(big_hat(X, Y))
% 275.77/40.20 = { by lemma 21 }
% 275.77/40.20 not(big_hat(Y, X))
% 275.77/40.20 = { by axiom 2 (big_hat_definition) }
% 275.77/40.20 not(not(big_V(not(Y), not(X))))
% 275.77/40.20 = { by lemma 18 }
% 275.77/40.20 big_V(not(Y), not(X))
% 275.77/40.20 = { by lemma 7 R->L }
% 275.77/40.20 big_V(not(X), not(Y))
% 275.77/40.20
% 275.77/40.20 Lemma 28: implies(not(X), not(implies(Y, X))) = big_V(X, Y).
% 275.77/40.20 Proof:
% 275.77/40.20 implies(not(X), not(implies(Y, X)))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(truth, implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by axiom 6 (wajsberg_2) R->L }
% 275.77/40.20 implies(implies(implies(implies(Y, X), X), implies(implies(X, X), implies(implies(Y, X), X))), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 implies(implies(big_V(Y, X), implies(implies(X, X), implies(implies(Y, X), X))), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by lemma 7 R->L }
% 275.77/40.20 implies(implies(big_V(X, Y), implies(implies(X, X), implies(implies(Y, X), X))), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by lemma 23 R->L }
% 275.77/40.20 implies(implies(big_V(X, Y), implies(implies(X, X), implies(not(X), not(implies(Y, X))))), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(implies(big_V(X, Y), implies(implies(implies(truth, X), X), implies(not(X), not(implies(Y, X))))), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 implies(implies(big_V(X, Y), implies(big_V(truth, X), implies(not(X), not(implies(Y, X))))), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by lemma 7 R->L }
% 275.77/40.20 implies(implies(big_V(X, Y), implies(big_V(X, truth), implies(not(X), not(implies(Y, X))))), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by lemma 9 }
% 275.77/40.20 implies(implies(big_V(X, Y), implies(truth, implies(not(X), not(implies(Y, X))))), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 implies(implies(big_V(X, Y), implies(not(X), not(implies(Y, X)))), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 big_V(big_V(X, Y), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 big_V(big_V(Y, X), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 big_V(implies(implies(Y, X), X), implies(not(X), not(implies(Y, X))))
% 275.77/40.20 = { by lemma 12 }
% 275.77/40.20 implies(implies(Y, X), X)
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 big_V(Y, X)
% 275.77/40.20 = { by lemma 7 R->L }
% 275.77/40.20 big_V(X, Y)
% 275.77/40.20
% 275.77/40.20 Lemma 29: big_V(X, implies(Y, X)) = implies(Y, X).
% 275.77/40.20 Proof:
% 275.77/40.20 big_V(X, implies(Y, X))
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(X, implies(Y, X)), implies(Y, X))
% 275.77/40.20 = { by lemma 13 }
% 275.77/40.20 implies(truth, implies(Y, X))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 implies(Y, X)
% 275.77/40.20
% 275.77/40.20 Lemma 30: implies(implies(big_V(X, Y), Z), implies(X, Z)) = truth.
% 275.77/40.20 Proof:
% 275.77/40.20 implies(implies(big_V(X, Y), Z), implies(X, Z))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(truth, implies(implies(big_V(X, Y), Z), implies(X, Z)))
% 275.77/40.20 = { by lemma 8 R->L }
% 275.77/40.20 implies(implies(X, big_V(X, Y)), implies(implies(big_V(X, Y), Z), implies(X, Z)))
% 275.77/40.20 = { by axiom 6 (wajsberg_2) }
% 275.77/40.20 truth
% 275.77/40.20
% 275.77/40.20 Lemma 31: big_V(implies(X, Y), implies(big_V(X, Z), Y)) = implies(X, Y).
% 275.77/40.20 Proof:
% 275.77/40.20 big_V(implies(X, Y), implies(big_V(X, Z), Y))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 big_V(implies(big_V(X, Z), Y), implies(X, Y))
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(implies(big_V(X, Z), Y), implies(X, Y)), implies(X, Y))
% 275.77/40.20 = { by lemma 30 }
% 275.77/40.20 implies(truth, implies(X, Y))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 implies(X, Y)
% 275.77/40.20
% 275.77/40.20 Lemma 32: big_V(X, big_V(X, Y)) = big_V(X, Y).
% 275.77/40.20 Proof:
% 275.77/40.20 big_V(X, big_V(X, Y))
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(X, big_V(X, Y)), big_V(X, Y))
% 275.77/40.20 = { by lemma 8 }
% 275.77/40.20 implies(truth, big_V(X, Y))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 big_V(X, Y)
% 275.77/40.20
% 275.77/40.20 Lemma 33: implies(implies(big_V(X, implies(Y, Z)), Z), big_V(Y, Z)) = truth.
% 275.77/40.20 Proof:
% 275.77/40.20 implies(implies(big_V(X, implies(Y, Z)), Z), big_V(Y, Z))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 implies(implies(big_V(implies(Y, Z), X), Z), big_V(Y, Z))
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(big_V(implies(Y, Z), X), Z), implies(implies(Y, Z), Z))
% 275.77/40.20 = { by lemma 30 }
% 275.77/40.20 truth
% 275.77/40.20
% 275.77/40.20 Lemma 34: big_V(X, big_hat(X, Y)) = X.
% 275.77/40.20 Proof:
% 275.77/40.20 big_V(X, big_hat(X, Y))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 big_V(big_hat(X, Y), X)
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(big_hat(X, Y), X), X)
% 275.77/40.20 = { by lemma 18 R->L }
% 275.77/40.20 implies(implies(big_hat(X, not(not(Y))), X), X)
% 275.77/40.20 = { by lemma 21 }
% 275.77/40.20 implies(implies(big_hat(not(not(Y)), X), X), X)
% 275.77/40.20 = { by lemma 18 R->L }
% 275.77/40.20 implies(implies(big_hat(not(not(Y)), not(not(X))), X), X)
% 275.77/40.20 = { by lemma 22 R->L }
% 275.77/40.20 implies(implies(big_hat(not(not(Y)), not(implies(X, not(truth)))), X), X)
% 275.77/40.20 = { by lemma 26 R->L }
% 275.77/40.20 implies(implies(not(big_V(not(Y), implies(X, not(truth)))), X), X)
% 275.77/40.20 = { by lemma 22 R->L }
% 275.77/40.20 implies(implies(implies(big_V(not(Y), implies(X, not(truth))), not(truth)), X), X)
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(implies(implies(big_V(not(Y), implies(X, not(truth))), not(truth)), implies(truth, X)), X)
% 275.77/40.20 = { by axiom 5 (wajsberg_4) R->L }
% 275.77/40.20 implies(implies(implies(big_V(not(Y), implies(X, not(truth))), not(truth)), implies(implies(implies(not(X), not(not(truth))), implies(not(truth), X)), X)), X)
% 275.77/40.20 = { by lemma 18 }
% 275.77/40.20 implies(implies(implies(big_V(not(Y), implies(X, not(truth))), not(truth)), implies(implies(implies(not(X), truth), implies(not(truth), X)), X)), X)
% 275.77/40.20 = { by lemma 10 }
% 275.77/40.20 implies(implies(implies(big_V(not(Y), implies(X, not(truth))), not(truth)), implies(implies(truth, implies(not(truth), X)), X)), X)
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 implies(implies(implies(big_V(not(Y), implies(X, not(truth))), not(truth)), implies(implies(not(truth), X), X)), X)
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 implies(implies(implies(big_V(not(Y), implies(X, not(truth))), not(truth)), big_V(not(truth), X)), X)
% 275.77/40.20 = { by lemma 7 R->L }
% 275.77/40.20 implies(implies(implies(big_V(not(Y), implies(X, not(truth))), not(truth)), big_V(X, not(truth))), X)
% 275.77/40.20 = { by lemma 33 }
% 275.77/40.20 implies(truth, X)
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 X
% 275.77/40.20
% 275.77/40.20 Lemma 35: implies(not(X), big_hat(not(X), not(Y))) = implies(Y, X).
% 275.77/40.20 Proof:
% 275.77/40.20 implies(not(X), big_hat(not(X), not(Y)))
% 275.77/40.20 = { by lemma 21 }
% 275.77/40.20 implies(not(X), big_hat(not(Y), not(X)))
% 275.77/40.20 = { by lemma 26 R->L }
% 275.77/40.20 implies(not(X), not(big_V(Y, X)))
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(not(X), not(implies(implies(Y, X), X)))
% 275.77/40.20 = { by lemma 28 }
% 275.77/40.20 big_V(X, implies(Y, X))
% 275.77/40.20 = { by lemma 29 }
% 275.77/40.20 implies(Y, X)
% 275.77/40.20
% 275.77/40.20 Lemma 36: implies(big_V(X, Y), not(implies(X, Y))) = not(Y).
% 275.77/40.20 Proof:
% 275.77/40.20 implies(big_V(X, Y), not(implies(X, Y)))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 implies(big_V(Y, X), not(implies(X, Y)))
% 275.77/40.20 = { by lemma 28 R->L }
% 275.77/40.20 implies(implies(not(Y), not(implies(X, Y))), not(implies(X, Y)))
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 big_V(not(Y), not(implies(X, Y)))
% 275.77/40.20 = { by lemma 34 R->L }
% 275.77/40.20 big_V(big_V(not(Y), big_hat(not(Y), not(X))), not(implies(X, Y)))
% 275.77/40.20 = { by lemma 35 R->L }
% 275.77/40.20 big_V(big_V(not(Y), big_hat(not(Y), not(X))), not(implies(not(Y), big_hat(not(Y), not(X)))))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 big_V(not(implies(not(Y), big_hat(not(Y), not(X)))), big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(not(implies(not(Y), big_hat(not(Y), not(X)))), big_V(not(Y), big_hat(not(Y), not(X)))), big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 implies(implies(not(implies(not(Y), big_hat(not(Y), not(X)))), big_V(big_hat(not(Y), not(X)), not(Y))), big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(implies(truth, implies(not(implies(not(Y), big_hat(not(Y), not(X)))), big_V(big_hat(not(Y), not(X)), not(Y)))), big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by axiom 5 (wajsberg_4) R->L }
% 275.77/40.20 implies(implies(implies(implies(not(big_hat(not(Y), not(X))), not(implies(not(Y), big_hat(not(Y), not(X))))), implies(implies(not(Y), big_hat(not(Y), not(X))), big_hat(not(Y), not(X)))), implies(not(implies(not(Y), big_hat(not(Y), not(X)))), big_V(big_hat(not(Y), not(X)), not(Y)))), big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by axiom 3 (big_V_definition) R->L }
% 275.77/40.20 implies(implies(implies(implies(not(big_hat(not(Y), not(X))), not(implies(not(Y), big_hat(not(Y), not(X))))), big_V(not(Y), big_hat(not(Y), not(X)))), implies(not(implies(not(Y), big_hat(not(Y), not(X)))), big_V(big_hat(not(Y), not(X)), not(Y)))), big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by lemma 7 R->L }
% 275.77/40.20 implies(implies(implies(implies(not(big_hat(not(Y), not(X))), not(implies(not(Y), big_hat(not(Y), not(X))))), big_V(big_hat(not(Y), not(X)), not(Y))), implies(not(implies(not(Y), big_hat(not(Y), not(X)))), big_V(big_hat(not(Y), not(X)), not(Y)))), big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 275.77/40.20 implies(implies(truth, implies(implies(implies(not(big_hat(not(Y), not(X))), not(implies(not(Y), big_hat(not(Y), not(X))))), big_V(big_hat(not(Y), not(X)), not(Y))), implies(not(implies(not(Y), big_hat(not(Y), not(X)))), big_V(big_hat(not(Y), not(X)), not(Y))))), big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by lemma 13 R->L }
% 275.77/40.20 implies(implies(implies(not(implies(not(Y), big_hat(not(Y), not(X)))), implies(not(big_hat(not(Y), not(X))), not(implies(not(Y), big_hat(not(Y), not(X)))))), implies(implies(implies(not(big_hat(not(Y), not(X))), not(implies(not(Y), big_hat(not(Y), not(X))))), big_V(big_hat(not(Y), not(X)), not(Y))), implies(not(implies(not(Y), big_hat(not(Y), not(X)))), big_V(big_hat(not(Y), not(X)), not(Y))))), big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by axiom 6 (wajsberg_2) }
% 275.77/40.20 implies(truth, big_V(not(Y), big_hat(not(Y), not(X))))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 big_V(not(Y), big_hat(not(Y), not(X)))
% 275.77/40.20 = { by lemma 34 }
% 275.77/40.20 not(Y)
% 275.77/40.20
% 275.77/40.20 Lemma 37: implies(X, not(implies(X, Y))) = big_V(not(X), not(Y)).
% 275.77/40.20 Proof:
% 275.77/40.20 implies(X, not(implies(X, Y)))
% 275.77/40.20 = { by lemma 18 R->L }
% 275.77/40.20 implies(X, not(implies(X, not(not(Y)))))
% 275.77/40.20 = { by lemma 34 R->L }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(X, not(not(Y)))))
% 275.77/40.20 = { by lemma 24 R->L }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(not(Y), not(X))))
% 275.77/40.20 = { by lemma 19 R->L }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(not(Y), big_V(not(X), not(X)))))
% 275.77/40.20 = { by lemma 35 R->L }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(not(big_V(not(X), not(X))), big_hat(not(big_V(not(X), not(X))), not(not(Y))))))
% 275.77/40.20 = { by axiom 2 (big_hat_definition) R->L }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(big_hat(X, X), big_hat(not(big_V(not(X), not(X))), not(not(Y))))))
% 275.77/40.20 = { by axiom 2 (big_hat_definition) R->L }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(big_hat(X, X), big_hat(big_hat(X, X), not(not(Y))))))
% 275.77/40.20 = { by lemma 21 R->L }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(big_hat(X, X), big_hat(not(not(Y)), big_hat(X, X)))))
% 275.77/40.20 = { by lemma 20 }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(not(not(X)), big_hat(not(not(Y)), big_hat(X, X)))))
% 275.77/40.20 = { by lemma 18 }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(X, big_hat(not(not(Y)), big_hat(X, X)))))
% 275.77/40.20 = { by lemma 20 }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(X, big_hat(not(not(Y)), not(not(X))))))
% 275.77/40.20 = { by lemma 18 }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(X, big_hat(not(not(Y)), X))))
% 275.77/40.20 = { by lemma 21 R->L }
% 275.77/40.20 implies(big_V(X, big_hat(X, not(not(Y)))), not(implies(X, big_hat(X, not(not(Y))))))
% 275.77/40.20 = { by lemma 36 }
% 275.77/40.20 not(big_hat(X, not(not(Y))))
% 275.77/40.20 = { by lemma 27 }
% 275.77/40.20 big_V(not(X), not(not(not(Y))))
% 275.77/40.20 = { by lemma 18 }
% 275.77/40.20 big_V(not(X), not(Y))
% 275.77/40.20
% 275.77/40.20 Lemma 38: big_V(implies(X, Y), implies(not(Y), big_hat(not(X), not(Z)))) = implies(X, Y).
% 275.77/40.20 Proof:
% 275.77/40.20 big_V(implies(X, Y), implies(not(Y), big_hat(not(X), not(Z))))
% 275.77/40.20 = { by lemma 26 R->L }
% 275.77/40.20 big_V(implies(X, Y), implies(not(Y), not(big_V(X, Z))))
% 275.77/40.20 = { by lemma 23 }
% 275.77/40.20 big_V(implies(X, Y), implies(big_V(X, Z), Y))
% 275.77/40.20 = { by lemma 7 }
% 275.77/40.20 big_V(implies(big_V(X, Z), Y), implies(X, Y))
% 275.77/40.20 = { by axiom 3 (big_V_definition) }
% 275.77/40.20 implies(implies(implies(big_V(X, Z), Y), implies(X, Y)), implies(X, Y))
% 275.77/40.20 = { by lemma 30 }
% 275.77/40.20 implies(truth, implies(X, Y))
% 275.77/40.20 = { by axiom 1 (wajsberg_1) }
% 275.77/40.20 implies(X, Y)
% 275.77/40.20
% 276.49/40.20 Lemma 39: big_V(X, big_V(Y, implies(Z, X))) = big_V(Y, implies(Z, X)).
% 276.49/40.20 Proof:
% 276.49/40.20 big_V(X, big_V(Y, implies(Z, X)))
% 276.49/40.20 = { by axiom 3 (big_V_definition) }
% 276.49/40.20 implies(implies(X, big_V(Y, implies(Z, X))), big_V(Y, implies(Z, X)))
% 276.49/40.20 = { by lemma 15 }
% 276.49/40.20 implies(truth, big_V(Y, implies(Z, X)))
% 276.49/40.20 = { by axiom 1 (wajsberg_1) }
% 276.49/40.20 big_V(Y, implies(Z, X))
% 276.49/40.20
% 276.49/40.20 Lemma 40: implies(big_V(X, Y), not(implies(Y, X))) = not(X).
% 276.49/40.20 Proof:
% 276.49/40.20 implies(big_V(X, Y), not(implies(Y, X)))
% 276.49/40.20 = { by lemma 7 }
% 276.49/40.20 implies(big_V(Y, X), not(implies(Y, X)))
% 276.49/40.20 = { by lemma 36 }
% 276.49/40.20 not(X)
% 276.49/40.20
% 276.49/40.20 Lemma 41: big_V(not(implies(X, Y)), big_V(X, Z)) = big_V(X, Z).
% 276.49/40.20 Proof:
% 276.49/40.20 big_V(not(implies(X, Y)), big_V(X, Z))
% 276.49/40.20 = { by lemma 18 R->L }
% 276.49/40.20 big_V(not(implies(X, Y)), big_V(X, not(not(Z))))
% 276.49/40.20 = { by lemma 18 R->L }
% 276.49/40.20 big_V(not(implies(X, Y)), big_V(not(not(X)), not(not(Z))))
% 276.49/40.20 = { by lemma 27 R->L }
% 276.49/40.20 big_V(not(implies(X, Y)), not(big_hat(not(X), not(Z))))
% 276.49/40.20 = { by lemma 37 R->L }
% 276.49/40.20 implies(implies(X, Y), not(implies(implies(X, Y), big_hat(not(X), not(Z)))))
% 276.49/40.20 = { by lemma 38 R->L }
% 276.49/40.20 implies(big_V(implies(X, Y), implies(not(Y), big_hat(not(X), not(Z)))), not(implies(implies(X, Y), big_hat(not(X), not(Z)))))
% 276.49/40.20 = { by lemma 39 R->L }
% 276.49/40.20 implies(big_V(big_hat(not(X), not(Z)), big_V(implies(X, Y), implies(not(Y), big_hat(not(X), not(Z))))), not(implies(implies(X, Y), big_hat(not(X), not(Z)))))
% 276.49/40.20 = { by lemma 38 }
% 276.49/40.20 implies(big_V(big_hat(not(X), not(Z)), implies(X, Y)), not(implies(implies(X, Y), big_hat(not(X), not(Z)))))
% 276.49/40.20 = { by lemma 40 }
% 276.49/40.20 not(big_hat(not(X), not(Z)))
% 276.49/40.20 = { by lemma 27 }
% 276.49/40.20 big_V(not(not(X)), not(not(Z)))
% 276.49/40.20 = { by lemma 18 }
% 276.49/40.20 big_V(X, not(not(Z)))
% 276.49/40.20 = { by lemma 18 }
% 276.49/40.20 big_V(X, Z)
% 276.49/40.20
% 276.49/40.20 Lemma 42: big_V(not(implies(X, Y)), big_V(Z, X)) = big_V(X, Z).
% 276.49/40.20 Proof:
% 276.49/40.20 big_V(not(implies(X, Y)), big_V(Z, X))
% 276.49/40.20 = { by lemma 7 }
% 276.49/40.20 big_V(not(implies(X, Y)), big_V(X, Z))
% 276.49/40.20 = { by lemma 41 }
% 276.49/40.20 big_V(X, Z)
% 276.49/40.20
% 276.49/40.20 Lemma 43: big_V(implies(X, implies(Y, Z)), implies(Y, implies(X, Z))) = implies(X, implies(Y, Z)).
% 276.49/40.20 Proof:
% 276.49/40.20 big_V(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 276.49/40.20 = { by lemma 28 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))))
% 276.49/40.20 = { by lemma 29 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(X, big_V(Z, implies(Y, Z))))))
% 276.49/40.20 = { by lemma 31 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), big_V(implies(X, big_V(Z, implies(Y, Z))), implies(big_V(X, Z), big_V(Z, implies(Y, Z)))))))
% 276.49/40.20 = { by lemma 32 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), big_V(implies(X, big_V(Z, implies(Y, Z))), implies(big_V(X, Z), big_V(Z, big_V(Z, implies(Y, Z))))))))
% 276.49/40.20 = { by lemma 7 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), big_V(implies(X, big_V(Z, implies(Y, Z))), implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))))))
% 276.49/40.20 = { by lemma 7 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), big_V(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z)), implies(X, big_V(Z, implies(Y, Z)))))))
% 276.49/40.20 = { by lemma 28 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(X, big_V(Z, implies(Y, Z))), implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))))))))
% 276.49/40.20 = { by lemma 7 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(X, big_V(Z, implies(Y, Z))), implies(big_V(X, Z), big_V(Z, big_V(Z, implies(Y, Z))))))))))
% 276.49/40.20 = { by lemma 7 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(X, big_V(Z, implies(Y, Z))), implies(big_V(Z, X), big_V(Z, big_V(Z, implies(Y, Z))))))))))
% 276.49/40.20 = { by lemma 28 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(X, big_V(Z, implies(Y, Z))), implies(big_V(Z, X), implies(not(Z), not(implies(big_V(Z, implies(Y, Z)), Z))))))))))
% 276.49/40.20 = { by lemma 28 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(not(Z), not(implies(X, Z))), implies(not(Z), not(implies(big_V(Z, implies(Y, Z)), Z))))))))))
% 276.49/40.20 = { by lemma 23 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(not(Z), not(implies(X, Z))), implies(implies(big_V(Z, implies(Y, Z)), Z), Z))))))))
% 276.49/40.20 = { by lemma 23 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(implies(X, Z), Z), implies(implies(big_V(Z, implies(Y, Z)), Z), Z))))))))
% 276.49/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(truth, implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(implies(X, Z), Z), implies(implies(big_V(Z, implies(Y, Z)), Z), Z)))))))))
% 276.49/40.20 = { by axiom 6 (wajsberg_2) R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(big_V(Z, implies(Y, Z)), Z), implies(X, Z))), implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(implies(X, Z), Z), implies(implies(big_V(Z, implies(Y, Z)), Z), Z)))))))))
% 276.49/40.20 = { by axiom 1 (wajsberg_1) R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(big_V(Z, implies(Y, Z)), Z), implies(X, Z))), implies(truth, implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(implies(X, Z), Z), implies(implies(big_V(Z, implies(Y, Z)), Z), Z))))))))))
% 276.49/40.20 = { by axiom 6 (wajsberg_2) R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(implies(implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(big_V(Z, implies(Y, Z)), Z), implies(X, Z))), implies(implies(implies(implies(big_V(Z, implies(Y, Z)), Z), implies(X, Z)), implies(implies(implies(X, Z), Z), implies(implies(big_V(Z, implies(Y, Z)), Z), Z))), implies(implies(X, big_V(Z, implies(Y, Z))), implies(implies(implies(X, Z), Z), implies(implies(big_V(Z, implies(Y, Z)), Z), Z))))))))))
% 276.49/40.20 = { by axiom 6 (wajsberg_2) }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z))), not(truth)))))
% 276.49/40.20 = { by lemma 22 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), not(not(implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z)))))))
% 276.49/40.20 = { by lemma 18 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(big_V(X, Z), big_V(big_V(Z, implies(Y, Z)), Z)))))
% 276.49/40.20 = { by lemma 7 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(big_V(X, Z), big_V(Z, big_V(Z, implies(Y, Z)))))))
% 276.49/40.20 = { by lemma 32 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(big_V(X, Z), big_V(Z, implies(Y, Z))))))
% 276.49/40.20 = { by lemma 29 }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(big_V(X, Z), implies(Y, Z)))))
% 276.49/40.20 = { by lemma 18 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(big_V(X, Z), implies(Y, not(not(Z)))))))
% 276.49/40.20 = { by lemma 42 R->L }
% 276.49/40.20 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, implies(X, Z)), implies(big_V(not(implies(X, Z)), big_V(Z, X)), implies(Y, not(not(Z)))))))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, not(not(implies(X, Z)))), implies(big_V(not(implies(X, Z)), big_V(Z, X)), implies(Y, not(not(Z)))))))
% 276.49/40.21 = { by lemma 40 R->L }
% 276.49/40.21 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, not(not(implies(X, Z)))), implies(big_V(not(implies(X, Z)), big_V(Z, X)), implies(Y, not(implies(big_V(Z, X), not(implies(X, Z)))))))))
% 276.49/40.21 = { by lemma 28 R->L }
% 276.49/40.21 implies(not(implies(X, implies(Y, Z))), not(implies(implies(Y, not(not(implies(X, Z)))), implies(implies(not(not(implies(X, Z))), not(implies(big_V(Z, X), not(implies(X, Z))))), implies(Y, not(implies(big_V(Z, X), not(implies(X, Z)))))))))
% 276.49/40.21 = { by axiom 6 (wajsberg_2) }
% 276.49/40.21 implies(not(implies(X, implies(Y, Z))), not(truth))
% 276.49/40.21 = { by lemma 22 }
% 276.49/40.21 not(not(implies(X, implies(Y, Z))))
% 276.49/40.21 = { by lemma 18 }
% 276.49/40.21 implies(X, implies(Y, Z))
% 276.49/40.21
% 276.49/40.21 Lemma 44: implies(Y, implies(X, Z)) = implies(X, implies(Y, Z)).
% 276.49/40.21 Proof:
% 276.49/40.21 implies(Y, implies(X, Z))
% 276.49/40.21 = { by lemma 43 R->L }
% 276.49/40.21 big_V(implies(Y, implies(X, Z)), implies(X, implies(Y, Z)))
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 big_V(implies(X, implies(Y, Z)), implies(Y, implies(X, Z)))
% 276.49/40.21 = { by lemma 43 }
% 276.49/40.21 implies(X, implies(Y, Z))
% 276.49/40.21
% 276.49/40.21 Lemma 45: implies(big_V(X, Y), big_V(X, big_V(Z, Y))) = truth.
% 276.49/40.21 Proof:
% 276.49/40.21 implies(big_V(X, Y), big_V(X, big_V(Z, Y)))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 implies(big_V(X, Y), big_V(X, big_V(Y, Z)))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 implies(big_V(X, Y), big_V(big_V(Y, Z), X))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 implies(big_V(Y, X), big_V(big_V(Y, Z), X))
% 276.49/40.21 = { by axiom 3 (big_V_definition) }
% 276.49/40.21 implies(implies(implies(Y, X), X), big_V(big_V(Y, Z), X))
% 276.49/40.21 = { by lemma 31 R->L }
% 276.49/40.21 implies(implies(big_V(implies(Y, X), implies(big_V(Y, Z), X)), X), big_V(big_V(Y, Z), X))
% 276.49/40.21 = { by lemma 33 }
% 276.49/40.21 truth
% 276.49/40.21
% 276.49/40.21 Lemma 46: big_V(big_V(X, Y), big_V(X, big_V(Y, Z))) = big_V(X, big_V(Y, Z)).
% 276.49/40.21 Proof:
% 276.49/40.21 big_V(big_V(X, Y), big_V(X, big_V(Y, Z)))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 big_V(big_V(X, Y), big_V(X, big_V(Z, Y)))
% 276.49/40.21 = { by axiom 3 (big_V_definition) }
% 276.49/40.21 implies(implies(big_V(X, Y), big_V(X, big_V(Z, Y))), big_V(X, big_V(Z, Y)))
% 276.49/40.21 = { by lemma 45 }
% 276.49/40.21 implies(truth, big_V(X, big_V(Z, Y)))
% 276.49/40.21 = { by axiom 1 (wajsberg_1) }
% 276.49/40.21 big_V(X, big_V(Z, Y))
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 big_V(X, big_V(Y, Z))
% 276.49/40.21
% 276.49/40.21 Lemma 47: big_V(big_V(X, Y), big_V(X, Z)) = big_V(Z, big_V(X, Y)).
% 276.49/40.21 Proof:
% 276.49/40.21 big_V(big_V(X, Y), big_V(X, Z))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 big_V(big_V(X, Y), big_V(Z, X))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 big_V(big_V(Y, X), big_V(Z, X))
% 276.49/40.21 = { by lemma 46 R->L }
% 276.49/40.21 big_V(big_V(big_V(Y, X), Z), big_V(big_V(Y, X), big_V(Z, X)))
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, X)), big_V(big_V(Y, X), big_V(Z, X)))
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, X)), big_V(big_V(Z, X), big_V(Y, X)))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, X)), big_V(big_V(Z, X), big_V(Y, not(not(X)))))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, X)), big_V(big_V(Z, X), big_V(not(not(Y)), not(not(X)))))
% 276.49/40.21 = { by lemma 27 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, X)), big_V(big_V(Z, X), not(big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, X)), big_V(big_V(Z, not(not(X))), not(big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, X)), big_V(big_V(not(not(Z)), not(not(X))), not(big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 27 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, X)), big_V(not(big_hat(not(Z), not(X))), not(big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 27 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, X)), not(big_hat(big_hat(not(Z), not(X)), big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(Y, not(not(X)))), not(big_hat(big_hat(not(Z), not(X)), big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 big_V(big_V(Z, big_V(not(not(Y)), not(not(X)))), not(big_hat(big_hat(not(Z), not(X)), big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 27 R->L }
% 276.49/40.21 big_V(big_V(Z, not(big_hat(not(Y), not(X)))), not(big_hat(big_hat(not(Z), not(X)), big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 big_V(big_V(not(not(Z)), not(big_hat(not(Y), not(X)))), not(big_hat(big_hat(not(Z), not(X)), big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 27 R->L }
% 276.49/40.21 big_V(not(big_hat(not(Z), big_hat(not(Y), not(X)))), not(big_hat(big_hat(not(Z), not(X)), big_hat(not(Y), not(X)))))
% 276.49/40.21 = { by lemma 37 R->L }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_hat(not(Z), big_hat(not(Y), not(X))), big_hat(big_hat(not(Z), not(X)), big_hat(not(Y), not(X))))))
% 276.49/40.21 = { by lemma 21 }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_hat(not(Z), big_hat(not(Y), not(X))), big_hat(big_hat(not(Y), not(X)), big_hat(not(Z), not(X))))))
% 276.49/40.21 = { by lemma 21 }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_hat(not(Z), big_hat(not(X), not(Y))), big_hat(big_hat(not(Y), not(X)), big_hat(not(Z), not(X))))))
% 276.49/40.21 = { by lemma 21 }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_hat(not(Z), big_hat(not(X), not(Y))), big_hat(big_hat(not(X), not(Y)), big_hat(not(Z), not(X))))))
% 276.49/40.21 = { by lemma 26 R->L }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_hat(not(Z), big_hat(not(X), not(Y))), big_hat(big_hat(not(X), not(Y)), not(big_V(Z, X))))))
% 276.49/40.21 = { by lemma 26 R->L }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_hat(not(Z), big_hat(not(X), not(Y))), big_hat(not(big_V(X, Y)), not(big_V(Z, X))))))
% 276.49/40.21 = { by lemma 26 R->L }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_hat(not(Z), big_hat(not(X), not(Y))), not(big_V(big_V(X, Y), big_V(Z, X))))))
% 276.49/40.21 = { by lemma 26 R->L }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_hat(not(Z), not(big_V(X, Y))), not(big_V(big_V(X, Y), big_V(Z, X))))))
% 276.49/40.21 = { by lemma 26 R->L }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(not(big_V(Z, big_V(X, Y))), not(big_V(big_V(X, Y), big_V(Z, X))))))
% 276.49/40.21 = { by lemma 23 }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_V(big_V(X, Y), big_V(Z, X)), big_V(Z, big_V(X, Y)))))
% 276.49/40.21 = { by lemma 46 R->L }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_V(big_V(X, Y), big_V(Z, X)), big_V(big_V(Z, X), big_V(Z, big_V(X, Y))))))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(implies(big_V(big_V(Z, X), big_V(X, Y)), big_V(big_V(Z, X), big_V(Z, big_V(X, Y))))))
% 276.49/40.21 = { by lemma 45 }
% 276.49/40.21 implies(big_hat(not(Z), big_hat(not(Y), not(X))), not(truth))
% 276.49/40.21 = { by lemma 22 }
% 276.49/40.21 not(big_hat(not(Z), big_hat(not(Y), not(X))))
% 276.49/40.21 = { by lemma 27 }
% 276.49/40.21 big_V(not(not(Z)), not(big_hat(not(Y), not(X))))
% 276.49/40.21 = { by lemma 18 }
% 276.49/40.21 big_V(Z, not(big_hat(not(Y), not(X))))
% 276.49/40.21 = { by lemma 27 }
% 276.49/40.21 big_V(Z, big_V(not(not(Y)), not(not(X))))
% 276.49/40.21 = { by lemma 18 }
% 276.49/40.21 big_V(Z, big_V(Y, not(not(X))))
% 276.49/40.21 = { by lemma 18 }
% 276.49/40.21 big_V(Z, big_V(Y, X))
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 big_V(Z, big_V(X, Y))
% 276.49/40.21
% 276.49/40.21 Lemma 48: big_V(big_V(X, Y), Z) = big_V(X, big_V(Z, Y)).
% 276.49/40.21 Proof:
% 276.49/40.21 big_V(big_V(X, Y), Z)
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 big_V(Z, big_V(X, Y))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 big_V(Z, big_V(Y, X))
% 276.49/40.21 = { by lemma 47 R->L }
% 276.49/40.21 big_V(big_V(Y, X), big_V(Y, Z))
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 big_V(big_V(Y, Z), big_V(Y, X))
% 276.49/40.21 = { by lemma 32 R->L }
% 276.49/40.21 big_V(big_V(Y, Z), big_V(big_V(Y, Z), big_V(Y, X)))
% 276.49/40.21 = { by lemma 47 }
% 276.49/40.21 big_V(big_V(Y, Z), big_V(X, big_V(Y, Z)))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 big_V(big_V(Y, Z), big_V(big_V(Y, Z), X))
% 276.49/40.21 = { by lemma 32 }
% 276.49/40.21 big_V(big_V(Y, Z), X)
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 big_V(X, big_V(Y, Z))
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 big_V(X, big_V(Z, Y))
% 276.49/40.21
% 276.49/40.21 Lemma 49: big_V(implies(X, Y), implies(implies(Z, X), Y)) = implies(X, Y).
% 276.49/40.21 Proof:
% 276.49/40.21 big_V(implies(X, Y), implies(implies(Z, X), Y))
% 276.49/40.21 = { by lemma 29 R->L }
% 276.49/40.21 big_V(implies(X, Y), implies(big_V(X, implies(Z, X)), Y))
% 276.49/40.21 = { by lemma 31 }
% 276.49/40.21 implies(X, Y)
% 276.49/40.21
% 276.49/40.21 Goal 1 (prove_mv_4): big_V(implies(x, y), implies(y, x)) = truth.
% 276.49/40.21 Proof:
% 276.49/40.21 big_V(implies(x, y), implies(y, x))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 big_V(implies(x, y), implies(not(not(y)), x))
% 276.49/40.21 = { by lemma 25 R->L }
% 276.49/40.21 big_V(implies(x, y), implies(not(x), not(y)))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 big_V(implies(not(x), not(y)), implies(x, y))
% 276.49/40.21 = { by lemma 28 R->L }
% 276.49/40.21 implies(not(implies(not(x), not(y))), not(implies(implies(x, y), implies(not(x), not(y)))))
% 276.49/40.21 = { by lemma 44 R->L }
% 276.49/40.21 implies(not(implies(not(x), not(y))), not(implies(not(x), implies(implies(x, y), not(y)))))
% 276.49/40.21 = { by lemma 25 R->L }
% 276.49/40.21 implies(not(not(implies(not(x), implies(implies(x, y), not(y))))), implies(not(x), not(y)))
% 276.49/40.21 = { by lemma 44 }
% 276.49/40.21 implies(not(x), implies(not(not(implies(not(x), implies(implies(x, y), not(y))))), not(y)))
% 276.49/40.21 = { by lemma 25 }
% 276.49/40.21 implies(not(x), implies(not(not(y)), not(implies(not(x), implies(implies(x, y), not(y))))))
% 276.49/40.21 = { by lemma 44 }
% 276.49/40.21 implies(not(not(y)), implies(not(x), not(implies(not(x), implies(implies(x, y), not(y))))))
% 276.49/40.21 = { by lemma 37 }
% 276.49/40.21 implies(not(not(y)), big_V(not(not(x)), not(implies(implies(x, y), not(y)))))
% 276.49/40.21 = { by lemma 18 }
% 276.49/40.21 implies(not(not(y)), big_V(x, not(implies(implies(x, y), not(y)))))
% 276.49/40.21 = { by lemma 18 }
% 276.49/40.21 implies(y, big_V(x, not(implies(implies(x, y), not(y)))))
% 276.49/40.21 = { by lemma 24 }
% 276.49/40.21 implies(y, big_V(x, not(implies(y, not(implies(x, y))))))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 implies(y, big_V(x, not(implies(y, not(implies(x, not(not(y))))))))
% 276.49/40.21 = { by lemma 18 R->L }
% 276.49/40.21 implies(y, big_V(x, not(implies(not(not(y)), not(implies(x, not(not(y))))))))
% 276.49/40.21 = { by lemma 23 }
% 276.49/40.21 implies(y, big_V(x, not(implies(implies(x, not(not(y))), not(y)))))
% 276.49/40.21 = { by lemma 42 R->L }
% 276.49/40.21 implies(y, big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(not(implies(implies(x, not(not(y))), not(y))), x)))
% 276.49/40.21 = { by lemma 28 R->L }
% 276.49/40.21 implies(y, implies(not(not(implies(x, not(implies(implies(x, not(not(y))), not(y)))))), not(implies(big_V(not(implies(implies(x, not(not(y))), not(y))), x), not(implies(x, not(implies(implies(x, not(not(y))), not(y)))))))))
% 276.49/40.21 = { by lemma 31 R->L }
% 276.49/40.21 implies(y, big_V(implies(not(not(implies(x, not(implies(implies(x, not(not(y))), not(y)))))), not(implies(big_V(not(implies(implies(x, not(not(y))), not(y))), x), not(implies(x, not(implies(implies(x, not(not(y))), not(y)))))))), implies(big_V(not(not(implies(x, not(implies(implies(x, not(not(y))), not(y)))))), implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y)))))), not(implies(big_V(not(implies(implies(x, not(not(y))), not(y))), x), not(implies(x, not(implies(implies(x, not(not(y))), not(y))))))))))
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 implies(y, big_V(implies(not(not(implies(x, not(implies(implies(x, not(not(y))), not(y)))))), not(implies(big_V(not(implies(implies(x, not(not(y))), not(y))), x), not(implies(x, not(implies(implies(x, not(not(y))), not(y)))))))), implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), not(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))))), not(implies(big_V(not(implies(implies(x, not(not(y))), not(y))), x), not(implies(x, not(implies(implies(x, not(not(y))), not(y))))))))))
% 276.49/40.21 = { by lemma 28 }
% 276.49/40.21 implies(y, big_V(big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(not(implies(implies(x, not(not(y))), not(y))), x)), implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), not(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))))), not(implies(big_V(not(implies(implies(x, not(not(y))), not(y))), x), not(implies(x, not(implies(implies(x, not(not(y))), not(y))))))))))
% 276.49/40.21 = { by lemma 48 }
% 276.49/40.21 implies(y, big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), not(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))))), not(implies(big_V(not(implies(implies(x, not(not(y))), not(y))), x), not(implies(x, not(implies(implies(x, not(not(y))), not(y)))))))), big_V(not(implies(implies(x, not(not(y))), not(y))), x))))
% 276.49/40.21 = { by lemma 7 R->L }
% 276.49/40.21 implies(y, big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(big_V(not(implies(implies(x, not(not(y))), not(y))), x), implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), not(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))))), not(implies(big_V(not(implies(implies(x, not(not(y))), not(y))), x), not(implies(x, not(implies(implies(x, not(not(y))), not(y)))))))))))
% 276.49/40.21 = { by lemma 40 }
% 276.49/40.21 implies(y, big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(big_V(not(implies(implies(x, not(not(y))), not(y))), x), implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), not(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))))), not(not(not(implies(implies(x, not(not(y))), not(y)))))))))
% 276.49/40.21 = { by lemma 48 }
% 276.49/40.21 implies(y, big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(not(implies(implies(x, not(not(y))), not(y))), big_V(implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), not(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))))), not(not(not(implies(implies(x, not(not(y))), not(y)))))), x))))
% 276.49/40.21 = { by lemma 18 }
% 276.49/40.21 implies(y, big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(not(implies(implies(x, not(not(y))), not(y))), big_V(implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y))))), not(not(not(implies(implies(x, not(not(y))), not(y)))))), x))))
% 276.49/40.21 = { by lemma 18 }
% 276.49/40.21 implies(y, big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(not(implies(implies(x, not(not(y))), not(y))), big_V(implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y))))), not(implies(implies(x, not(not(y))), not(y)))), x))))
% 276.49/40.21 = { by lemma 7 }
% 276.49/40.21 implies(y, big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(not(implies(implies(x, not(not(y))), not(y))), big_V(x, implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y))))), not(implies(implies(x, not(not(y))), not(y))))))))
% 276.49/40.21 = { by lemma 39 }
% 276.49/40.22 implies(y, big_V(not(implies(x, not(implies(implies(x, not(not(y))), not(y))))), big_V(x, implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y))))), not(implies(implies(x, not(not(y))), not(y)))))))
% 276.49/40.22 = { by lemma 41 }
% 276.49/40.22 implies(y, big_V(x, implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y))))), not(implies(implies(x, not(not(y))), not(y))))))
% 276.49/40.22 = { by lemma 23 R->L }
% 276.49/40.22 implies(y, big_V(x, implies(not(not(implies(implies(x, not(not(y))), not(y)))), not(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y)))))))))
% 276.49/40.22 = { by lemma 22 R->L }
% 276.49/40.22 implies(y, big_V(x, implies(not(not(implies(implies(x, not(not(y))), not(y)))), implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y))))), not(truth)))))
% 276.49/40.22 = { by axiom 6 (wajsberg_2) R->L }
% 276.49/40.22 implies(y, big_V(x, implies(not(not(implies(implies(x, not(not(y))), not(y)))), implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y))))), not(implies(implies(x, not(implies(implies(x, not(not(y))), not(y)))), implies(implies(not(implies(implies(x, not(not(y))), not(y))), implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))))))))))
% 276.49/40.22 = { by lemma 13 }
% 276.49/40.22 implies(y, big_V(x, implies(not(not(implies(implies(x, not(not(y))), not(y)))), implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y))))), not(implies(implies(x, not(implies(implies(x, not(not(y))), not(y)))), implies(truth, implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))))))))))
% 276.49/40.22 = { by axiom 1 (wajsberg_1) }
% 276.49/40.22 implies(y, big_V(x, implies(not(not(implies(implies(x, not(not(y))), not(y)))), implies(big_V(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y))))), implies(x, not(implies(implies(x, not(not(y))), not(y))))), not(implies(implies(x, not(implies(implies(x, not(not(y))), not(y)))), implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y)))))))))))
% 276.49/40.22 = { by lemma 40 }
% 276.49/40.22 implies(y, big_V(x, implies(not(not(implies(implies(x, not(not(y))), not(y)))), not(implies(x, implies(big_V(not(y), implies(x, not(not(y)))), not(implies(implies(x, not(not(y))), not(y)))))))))
% 276.49/40.22 = { by lemma 40 }
% 276.49/40.22 implies(y, big_V(x, implies(not(not(implies(implies(x, not(not(y))), not(y)))), not(implies(x, not(not(y)))))))
% 276.49/40.22 = { by lemma 18 }
% 276.49/40.22 implies(y, big_V(x, implies(implies(implies(x, not(not(y))), not(y)), not(implies(x, not(not(y)))))))
% 276.49/40.22 = { by lemma 24 }
% 276.49/40.22 implies(y, big_V(x, implies(implies(x, not(not(y))), not(implies(implies(x, not(not(y))), not(y))))))
% 276.49/40.22 = { by lemma 37 }
% 276.49/40.22 implies(y, big_V(x, big_V(not(implies(x, not(not(y)))), not(not(y)))))
% 276.49/40.22 = { by lemma 7 }
% 276.49/40.22 implies(y, big_V(x, big_V(not(not(y)), not(implies(x, not(not(y)))))))
% 276.49/40.22 = { by lemma 18 R->L }
% 276.49/40.22 implies(y, big_V(x, big_V(not(not(y)), not(implies(x, not(not(not(not(y)))))))))
% 276.49/40.22 = { by lemma 24 R->L }
% 276.49/40.22 implies(y, big_V(x, big_V(not(not(y)), not(implies(not(not(not(y))), not(x))))))
% 276.49/40.22 = { by lemma 7 }
% 276.49/40.22 implies(y, big_V(x, big_V(not(implies(not(not(not(y))), not(x))), not(not(y)))))
% 276.49/40.22 = { by lemma 48 R->L }
% 276.49/40.22 implies(y, big_V(big_V(x, not(not(y))), not(implies(not(not(not(y))), not(x)))))
% 276.49/40.22 = { by lemma 28 R->L }
% 276.49/40.22 implies(y, big_V(implies(not(x), not(implies(not(not(y)), x))), not(implies(not(not(not(y))), not(x)))))
% 276.49/40.22 = { by lemma 7 }
% 276.49/40.22 implies(y, big_V(not(implies(not(not(not(y))), not(x))), implies(not(x), not(implies(not(not(y)), x)))))
% 276.49/40.22 = { by lemma 49 R->L }
% 276.49/40.22 implies(y, big_V(not(implies(not(not(not(y))), not(x))), big_V(implies(not(x), not(implies(not(not(y)), x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))))))
% 276.49/40.22 = { by lemma 7 }
% 276.49/40.22 implies(y, big_V(not(implies(not(not(not(y))), not(x))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x))))))
% 276.49/40.22 = { by axiom 1 (wajsberg_1) R->L }
% 276.49/40.22 implies(y, big_V(not(implies(not(not(not(y))), not(x))), implies(truth, big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))))
% 276.49/40.22 = { by lemma 16 R->L }
% 276.49/40.22 implies(y, big_V(not(implies(not(not(not(y))), not(x))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))))
% 276.49/40.22 = { by axiom 3 (big_V_definition) }
% 276.49/40.22 implies(y, implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))))
% 276.49/40.22 = { by axiom 1 (wajsberg_1) R->L }
% 276.49/40.22 implies(y, implies(implies(truth, implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x))))))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))))
% 276.49/40.22 = { by lemma 14 R->L }
% 276.49/40.22 implies(y, implies(implies(implies(implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))), implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x))))))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))))
% 276.49/40.22 = { by axiom 3 (big_V_definition) R->L }
% 276.49/40.22 implies(y, implies(implies(implies(big_V(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))), implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x))))))), implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))))
% 276.49/40.22 = { by lemma 30 }
% 276.49/40.22 implies(y, implies(truth, implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))))
% 276.49/40.22 = { by axiom 1 (wajsberg_1) }
% 276.49/40.22 implies(y, implies(implies(not(implies(not(not(not(y))), not(x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))), big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x))))))
% 276.49/40.22 = { by lemma 16 }
% 276.49/40.22 implies(y, implies(truth, big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x))))))
% 276.49/40.22 = { by axiom 1 (wajsberg_1) }
% 276.49/40.22 implies(y, big_V(implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x))), implies(not(x), not(implies(not(not(y)), x)))))
% 276.49/40.22 = { by lemma 7 R->L }
% 276.49/40.22 implies(y, big_V(implies(not(x), not(implies(not(not(y)), x))), implies(implies(not(not(not(y))), not(x)), not(implies(not(not(y)), x)))))
% 276.49/40.22 = { by lemma 49 }
% 276.49/40.22 implies(y, implies(not(x), not(implies(not(not(y)), x))))
% 276.49/40.22 = { by lemma 28 }
% 276.49/40.22 implies(y, big_V(x, not(not(y))))
% 276.49/40.22 = { by lemma 18 }
% 276.49/40.22 implies(y, big_V(x, y))
% 276.49/40.22 = { by lemma 7 }
% 276.49/40.22 implies(y, big_V(y, x))
% 276.49/40.22 = { by lemma 8 }
% 276.49/40.22 truth
% 276.49/40.22 % SZS output end Proof
% 276.49/40.22
% 276.49/40.22 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------