%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : LCL198-3 : TPTP v9.3.1. Released v2.3.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n009.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 11:49:18 AM UTC 2026
% Result : Unsatisfiable 22.18s 3.34s
% Output : Proof 26.16s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.04 % Problem : LCL198-3 : TPTP v9.3.1. Released v2.3.0.
% 0.00/0.06 % Command : run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.16/0.42 % Computer : n009.cluster.edu
% 0.16/0.42 % Model : x86_64 x86_64
% 0.16/0.42 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.16/0.42 % Memory : 8046.5625MB
% 0.16/0.42 % OS : Linux 6.8.0-71-generic
% 0.16/0.42 % CPULimit : 300
% 0.16/0.42 % WCLimit : 300
% 0.16/0.42 % DateTime : Sun Sep 27 15:25:59 UTC 2026
% 0.16/0.42 % CPUTime :
% 0.16/0.42 Running run_twee /export/starexec/sandbox/benchmark/theBenchmark.p
% 22.18/3.34 Command-line arguments: --lhs-weight 1 --flip-ordering --normalise-queue-percent 10 --cp-renormalise-threshold 10 --complete-subsets --ground-joining-incomplete-limit 15 --flatten-every 2
% 22.18/3.34
% 22.18/3.34 % SZS status Unsatisfiable
% 22.18/3.34
% 23.78/3.59 % SZS output start Proof
% 23.78/3.59 Axiom 1 (implies_definition): implies(X, Y) = or(not(X), Y).
% 23.78/3.59 Axiom 2 (ifeq_axiom): ifeq(X, X, Y, Z) = Y.
% 23.78/3.59 Axiom 3 (axiom_1_3): axiom(implies(X, or(Y, X))) = true.
% 23.78/3.59 Axiom 4 (axiom_1_2): axiom(implies(or(X, X), X)) = true.
% 23.78/3.59 Axiom 5 (rule_1): ifeq(axiom(X), true, theorem(X), true) = true.
% 23.78/3.59 Axiom 6 (axiom_1_4): axiom(implies(or(X, Y), or(Y, X))) = true.
% 23.78/3.59 Axiom 7 (rule_2): ifeq(theorem(implies(X, Y)), true, ifeq(theorem(X), true, theorem(Y), true), true) = true.
% 23.78/3.59 Axiom 8 (axiom_1_6): axiom(implies(implies(X, Y), implies(or(Z, X), or(Z, Y)))) = true.
% 23.78/3.60 Axiom 9 (axiom_1_5): axiom(implies(or(X, or(Y, Z)), or(Y, or(X, Z)))) = true.
% 23.78/3.60
% 23.78/3.60 Goal 1 (prove_this): theorem(implies(or(not(p), implies(p, q)), implies(p, q))) = true.
% 23.78/3.60 Proof:
% 23.78/3.60 theorem(implies(or(not(p), implies(p, q)), implies(p, q)))
% 24.58/3.60 = { by axiom 2 (ifeq_axiom) R->L }
% 24.58/3.60 ifeq(true, true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.60 = { by axiom 7 (rule_2) R->L }
% 24.58/3.60 ifeq(ifeq(theorem(implies(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)))), true, ifeq(theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.60 = { by axiom 2 (ifeq_axiom) R->L }
% 24.58/3.60 ifeq(ifeq(ifeq(true, true, theorem(implies(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)))), true), true, ifeq(theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.60 = { by axiom 9 (axiom_1_5) R->L }
% 24.58/3.61 ifeq(ifeq(ifeq(axiom(implies(or(not(implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), or(not(implies(p, q)), q)))), true, theorem(implies(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)))), true), true, ifeq(theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.61 = { by axiom 1 (implies_definition) R->L }
% 24.58/3.61 ifeq(ifeq(ifeq(axiom(implies(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), or(not(implies(p, q)), q)))), true, theorem(implies(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)))), true), true, ifeq(theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.61 = { by axiom 1 (implies_definition) R->L }
% 24.58/3.61 ifeq(ifeq(ifeq(axiom(implies(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)))), true, theorem(implies(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)))), true), true, ifeq(theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.61 = { by axiom 5 (rule_1) }
% 24.58/3.61 ifeq(ifeq(true, true, ifeq(theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.61 = { by axiom 2 (ifeq_axiom) }
% 24.58/3.61 ifeq(ifeq(theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.61 = { by axiom 2 (ifeq_axiom) R->L }
% 24.58/3.61 ifeq(ifeq(ifeq(true, true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.62 = { by axiom 7 (rule_2) R->L }
% 24.58/3.62 ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, ifeq(theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.62 = { by axiom 2 (ifeq_axiom) R->L }
% 24.58/3.62 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true), true, ifeq(theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.62 = { by axiom 8 (axiom_1_6) R->L }
% 24.58/3.63 ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(or(not(implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), or(not(implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true), true, ifeq(theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.63 = { by axiom 1 (implies_definition) R->L }
% 24.58/3.63 ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), or(not(implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true), true, ifeq(theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.63 = { by axiom 1 (implies_definition) R->L }
% 24.58/3.63 ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true), true, ifeq(theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.64 = { by axiom 5 (rule_1) }
% 24.58/3.64 ifeq(ifeq(ifeq(ifeq(true, true, ifeq(theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.64 = { by axiom 2 (ifeq_axiom) }
% 24.58/3.64 ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.65 = { by axiom 2 (ifeq_axiom) R->L }
% 24.58/3.65 ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.65 = { by axiom 7 (rule_2) R->L }
% 24.58/3.65 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, ifeq(theorem(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.66 = { by axiom 2 (ifeq_axiom) R->L }
% 24.58/3.66 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, ifeq(ifeq(true, true, theorem(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.66 = { by axiom 6 (axiom_1_4) R->L }
% 24.58/3.66 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, ifeq(ifeq(axiom(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true, theorem(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.67 = { by axiom 5 (rule_1) }
% 24.58/3.67 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, ifeq(true, true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.67 = { by axiom 2 (ifeq_axiom) }
% 24.58/3.67 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.68 = { by axiom 2 (ifeq_axiom) R->L }
% 24.58/3.68 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.68 = { by axiom 8 (axiom_1_6) R->L }
% 24.58/3.69 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(or(not(implies(or(not(p), implies(p, q)), implies(p, q))), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), or(not(implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, theorem(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.69 = { by axiom 1 (implies_definition) R->L }
% 24.58/3.69 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), or(not(implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, theorem(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.69 = { by axiom 1 (implies_definition) R->L }
% 24.58/3.70 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true, theorem(implies(implies(or(q, implies(or(not(p), implies(p, q)), implies(p, q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)), implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))))), true), true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 24.58/3.70 = { by axiom 5 (rule_1) }
% 24.58/3.70 ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.70 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.70 ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.70 = { by axiom 2 (ifeq_axiom) R->L }
% 25.34/3.71 ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, ifeq(true, true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.71 = { by axiom 5 (rule_1) R->L }
% 25.34/3.71 ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, ifeq(ifeq(axiom(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.71 = { by axiom 3 (axiom_1_3) }
% 25.34/3.71 ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, ifeq(ifeq(true, true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.71 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.71 ifeq(ifeq(ifeq(ifeq(ifeq(theorem(implies(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q)))), implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, ifeq(theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(q, implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(implies(or(not(p), implies(p, q)), implies(p, q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true), true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.71 = { by axiom 7 (rule_2) }
% 25.34/3.71 ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.71 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.72 ifeq(ifeq(ifeq(theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.72 = { by axiom 2 (ifeq_axiom) R->L }
% 25.34/3.72 ifeq(ifeq(ifeq(theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, ifeq(true, true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.72 = { by axiom 5 (rule_1) R->L }
% 25.34/3.72 ifeq(ifeq(ifeq(theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, ifeq(ifeq(axiom(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.72 = { by axiom 1 (implies_definition) }
% 25.34/3.72 ifeq(ifeq(ifeq(theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, ifeq(ifeq(axiom(implies(implies(p, q), or(not(or(not(p), implies(p, q))), implies(p, q)))), true, theorem(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.72 = { by axiom 3 (axiom_1_3) }
% 25.34/3.72 ifeq(ifeq(ifeq(theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, ifeq(ifeq(true, true, theorem(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.72 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.72 ifeq(ifeq(ifeq(theorem(implies(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q)))), true, ifeq(theorem(implies(implies(p, q), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(implies(p, q), or(implies(or(not(p), implies(p, q)), implies(p, q)), q))), true), true), true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.72 = { by axiom 7 (rule_2) }
% 25.34/3.73 ifeq(ifeq(true, true, theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.73 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.73 ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true)
% 25.34/3.73 = { by axiom 2 (ifeq_axiom) R->L }
% 25.34/3.73 ifeq(true, true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.73 = { by axiom 7 (rule_2) R->L }
% 25.34/3.73 ifeq(ifeq(theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.73 = { by axiom 2 (ifeq_axiom) R->L }
% 25.34/3.73 ifeq(ifeq(ifeq(true, true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.74 = { by axiom 5 (rule_1) R->L }
% 25.34/3.74 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.74 = { by axiom 1 (implies_definition) }
% 25.34/3.74 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), or(not(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.75 = { by axiom 1 (implies_definition) }
% 25.34/3.75 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), or(not(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.75 = { by axiom 8 (axiom_1_6) }
% 25.34/3.75 ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.76 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.76 ifeq(ifeq(ifeq(theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.76 = { by axiom 2 (ifeq_axiom) R->L }
% 25.34/3.76 ifeq(ifeq(ifeq(theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, ifeq(true, true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.77 = { by axiom 5 (rule_1) R->L }
% 25.34/3.77 ifeq(ifeq(ifeq(theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, ifeq(ifeq(axiom(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.77 = { by axiom 4 (axiom_1_2) }
% 25.34/3.77 ifeq(ifeq(ifeq(theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, ifeq(ifeq(true, true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.78 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.78 ifeq(ifeq(ifeq(theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q))), implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true), true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.78 = { by axiom 7 (rule_2) }
% 25.34/3.78 ifeq(ifeq(true, true, ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.78 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.78 ifeq(ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.78 = { by axiom 2 (ifeq_axiom) R->L }
% 25.34/3.78 ifeq(ifeq(ifeq(true, true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.78 = { by axiom 5 (rule_1) R->L }
% 25.34/3.78 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.78 = { by axiom 1 (implies_definition) R->L }
% 25.34/3.79 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(implies(p, q), q), implies(implies(p, implies(p, q)), implies(p, q)))), true, theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.79 = { by axiom 1 (implies_definition) }
% 25.34/3.79 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(implies(p, q), q), implies(implies(p, implies(p, q)), or(not(p), q)))), true, theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.79 = { by axiom 1 (implies_definition) }
% 25.34/3.79 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), or(not(p), q)))), true, theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.79 = { by axiom 8 (axiom_1_6) }
% 25.34/3.79 ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.79 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.79 ifeq(ifeq(ifeq(theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.79 = { by axiom 2 (ifeq_axiom) R->L }
% 25.34/3.79 ifeq(ifeq(ifeq(true, true, ifeq(theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.79 = { by axiom 5 (rule_1) R->L }
% 25.34/3.80 ifeq(ifeq(ifeq(ifeq(axiom(implies(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, theorem(implies(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true), true, ifeq(theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.80 = { by axiom 8 (axiom_1_6) }
% 25.34/3.80 ifeq(ifeq(ifeq(ifeq(true, true, theorem(implies(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true), true, ifeq(theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.80 = { by axiom 2 (ifeq_axiom) }
% 25.34/3.80 ifeq(ifeq(ifeq(theorem(implies(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q))), implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q)))))), true, ifeq(theorem(implies(implies(implies(p, q), q), implies(or(not(p), implies(p, q)), implies(p, q)))), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(or(not(p), implies(p, q)), implies(p, q))))), true), true), true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 25.34/3.80 = { by axiom 7 (rule_2) }
% 25.34/3.80 ifeq(ifeq(true, true, theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 26.16/3.80 = { by axiom 2 (ifeq_axiom) }
% 26.16/3.80 ifeq(theorem(implies(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q)), implies(or(not(p), implies(p, q)), implies(p, q)))), true, ifeq(theorem(or(implies(or(not(p), implies(p, q)), implies(p, q)), implies(implies(p, q), q))), true, theorem(implies(or(not(p), implies(p, q)), implies(p, q))), true), true)
% 26.16/3.80 = { by axiom 7 (rule_2) }
% 26.16/3.80 true
% 26.16/3.80 % SZS output end Proof
% 26.16/3.80
% 26.16/3.80 RESULT: Unsatisfiable (the axioms are contradictory).
%------------------------------------------------------------------------------