0.04/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.04/0.12	% Command    : twee %s --tstp --casc --quiet --explain-encoding --conditional-encoding if --smaller --drop-non-horn
0.12/0.33	% Computer   : n016.cluster.edu
0.12/0.33	% Model      : x86_64 x86_64
0.12/0.33	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.12/0.33	% Memory     : 8042.1875MB
0.12/0.33	% OS         : Linux 3.10.0-693.el7.x86_64
0.12/0.33	% CPULimit   : 180
0.12/0.33	% DateTime   : Thu Aug 29 13:07:23 EDT 2019
0.12/0.33	% CPUTime    : 
6.33/6.52	% SZS status Unsatisfiable
6.33/6.52	
6.33/6.52	% SZS output start Proof
6.33/6.52	Take the following subset of the input axioms:
7.09/7.27	  fof(cn_1, axiom, ![X, Y, Z]: is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z))))=true).
7.09/7.27	  fof(cn_2, axiom, ![X]: is_a_theorem(implies(implies(not(X), X), X))=true).
7.09/7.27	  fof(cn_3, axiom, ![X, Y]: true=is_a_theorem(implies(X, implies(not(X), Y)))).
7.09/7.27	  fof(condensed_detachment, axiom, ![X, Y]: true=ifeq(is_a_theorem(implies(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true)).
7.09/7.27	  fof(ifeq_axiom, axiom, ![B, A, C]: B=ifeq(A, A, B, C)).
7.09/7.27	  fof(prove_cn_21, negated_conjecture, true!=is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c))))).
7.09/7.27	
7.09/7.27	Now clausify the problem and encode Horn clauses using encoding 3 of
7.09/7.27	http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
7.09/7.27	We repeatedly replace C & s=t => u=v by the two clauses:
7.09/7.27	  fresh(y, y, x1...xn) = u
7.09/7.27	  C => fresh(s, t, x1...xn) = v
7.09/7.27	where fresh is a fresh function symbol and x1..xn are the free
7.09/7.27	variables of u and v.
7.09/7.27	A predicate p(X) is encoded as p(X)=true (this is sound, because the
7.09/7.27	input problem has no model of domain size 1).
7.09/7.27	
7.09/7.27	The encoding turns the above axioms into the following unit equations and goals:
7.09/7.27	
7.09/7.27	Axiom 1 (cn_2): is_a_theorem(implies(implies(not(X), X), X)) = true.
7.09/7.27	Axiom 2 (condensed_detachment): true = ifeq(is_a_theorem(implies(X, Y)), true, ifeq(is_a_theorem(X), true, is_a_theorem(Y), true), true).
7.09/7.27	Axiom 3 (ifeq_axiom): X = ifeq(Y, Y, X, Z).
7.09/7.27	Axiom 4 (cn_3): true = is_a_theorem(implies(X, implies(not(X), Y))).
7.09/7.27	Axiom 5 (cn_1): is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))) = true.
7.09/7.27	
7.09/7.27	Lemma 6: ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true) = true.
7.09/7.27	Proof:
7.09/7.27	  ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true)
7.09/7.27	= { by axiom 3 (ifeq_axiom) }
7.09/7.27	  ifeq(true, true, ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true), true)
7.09/7.27	= { by axiom 5 (cn_1) }
7.09/7.27	  ifeq(is_a_theorem(implies(implies(X, Y), implies(implies(Y, Z), implies(X, Z)))), true, ifeq(is_a_theorem(implies(X, Y)), true, is_a_theorem(implies(implies(Y, Z), implies(X, Z))), true), true)
7.09/7.27	= { by axiom 2 (condensed_detachment) }
7.09/7.27	  true
7.09/7.27	
7.09/7.27	Lemma 7: is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))) = true.
7.09/7.27	Proof:
7.09/7.27	  is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z)))
7.09/7.27	= { by axiom 3 (ifeq_axiom) }
7.09/7.27	  ifeq(true, true, is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true)
7.09/7.27	= { by axiom 5 (cn_1) }
7.09/7.27	  ifeq(is_a_theorem(implies(implies(X, implies(not(X), Y)), implies(implies(implies(not(X), Y), Z), implies(X, Z)))), true, is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true)
7.09/7.27	= { by axiom 3 (ifeq_axiom) }
7.09/7.27	  ifeq(is_a_theorem(implies(implies(X, implies(not(X), Y)), implies(implies(implies(not(X), Y), Z), implies(X, Z)))), true, ifeq(true, true, is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true), true)
7.09/7.27	= { by axiom 4 (cn_3) }
7.09/7.27	  ifeq(is_a_theorem(implies(implies(X, implies(not(X), Y)), implies(implies(implies(not(X), Y), Z), implies(X, Z)))), true, ifeq(is_a_theorem(implies(X, implies(not(X), Y))), true, is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true), true)
7.09/7.27	= { by axiom 2 (condensed_detachment) }
7.09/7.27	  true
7.09/7.27	
7.09/7.27	Lemma 8: ifeq(is_a_theorem(implies(implies(not(X), Y), Z)), true, is_a_theorem(implies(X, Z)), true) = true.
7.09/7.27	Proof:
7.09/7.27	  ifeq(is_a_theorem(implies(implies(not(X), Y), Z)), true, is_a_theorem(implies(X, Z)), true)
7.09/7.27	= { by axiom 3 (ifeq_axiom) }
7.09/7.27	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(X), Y), Z)), true, is_a_theorem(implies(X, Z)), true), true)
7.09/7.27	= { by lemma 7 }
7.09/7.27	  ifeq(is_a_theorem(implies(implies(implies(not(X), Y), Z), implies(X, Z))), true, ifeq(is_a_theorem(implies(implies(not(X), Y), Z)), true, is_a_theorem(implies(X, Z)), true), true)
7.09/7.27	= { by axiom 2 (condensed_detachment) }
7.09/7.27	  true
7.09/7.27	
7.09/7.27	Lemma 9: is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))) = true.
7.09/7.27	Proof:
7.09/7.27	  is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W)))
7.09/7.27	= { by axiom 3 (ifeq_axiom) }
7.09/7.27	  ifeq(true, true, is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true)
7.09/7.27	= { by axiom 5 (cn_1) }
7.09/7.27	  ifeq(is_a_theorem(implies(implies(Z, X), implies(implies(X, Y), implies(Z, Y)))), true, is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true)
7.09/7.27	= { by lemma 6 }
7.09/7.27	  true
7.09/7.27	
7.09/7.27	Lemma 10: ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true) = true.
7.09/7.27	Proof:
7.09/7.27	  ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true)
7.09/7.27	= { by axiom 3 (ifeq_axiom) }
7.09/7.27	  ifeq(true, true, ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true), true)
7.09/7.27	= { by lemma 9 }
7.09/7.27	  ifeq(is_a_theorem(implies(implies(implies(implies(X, Y), implies(Z, Y)), W), implies(implies(Z, X), W))), true, ifeq(is_a_theorem(implies(implies(implies(X, Y), implies(Z, Y)), W)), true, is_a_theorem(implies(implies(Z, X), W)), true), true)
7.09/7.27	= { by axiom 2 (condensed_detachment) }
7.09/7.27	  true
7.09/7.27	
7.09/7.27	Lemma 11: is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))) = true.
7.09/7.27	Proof:
7.09/7.27	  is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y)))
7.09/7.27	= { by axiom 3 (ifeq_axiom) }
7.09/7.27	  ifeq(true, true, is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true)
7.09/7.27	= { by axiom 1 (cn_2) }
7.09/7.27	  ifeq(is_a_theorem(implies(implies(not(X), X), X)), true, is_a_theorem(implies(implies(X, Y), implies(implies(not(X), X), Y))), true)
7.09/7.27	= { by lemma 6 }
7.86/8.04	  true
7.86/8.04	
7.86/8.04	Goal 1 (prove_cn_21): true = is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))).
7.86/8.04	Proof:
7.86/8.04	  true
7.86/8.04	= { by axiom 2 (condensed_detachment) }
7.86/8.04	  ifeq(is_a_theorem(implies(implies(b, implies(implies(b, c), c)), implies(implies(a, implies(b, c)), implies(b, implies(a, c))))), true, ifeq(is_a_theorem(implies(b, implies(implies(b, c), c))), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true), true)
7.86/8.04	= { by axiom 3 (ifeq_axiom) }
7.86/8.04	  ifeq(ifeq(true, true, is_a_theorem(implies(implies(b, implies(implies(b, c), c)), implies(implies(a, implies(b, c)), implies(b, implies(a, c))))), true), true, ifeq(is_a_theorem(implies(b, implies(implies(b, c), c))), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true), true)
7.86/8.04	= { by lemma 9 }
7.86/8.04	  ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(b, c), c), implies(a, c)), implies(b, implies(a, c))), implies(implies(a, implies(b, c)), implies(b, implies(a, c))))), true, is_a_theorem(implies(implies(b, implies(implies(b, c), c)), implies(implies(a, implies(b, c)), implies(b, implies(a, c))))), true), true, ifeq(is_a_theorem(implies(b, implies(implies(b, c), c))), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true), true)
7.86/8.04	= { by lemma 10 }
7.86/8.04	  ifeq(true, true, ifeq(is_a_theorem(implies(b, implies(implies(b, c), c))), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true), true)
7.86/8.04	= { by axiom 3 (ifeq_axiom) }
7.86/8.04	  ifeq(is_a_theorem(implies(b, implies(implies(b, c), c))), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by axiom 3 (ifeq_axiom) }
7.86/8.04	  ifeq(ifeq(true, true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by axiom 2 (condensed_detachment) }
7.86/8.04	  ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by axiom 3 (ifeq_axiom) }
7.86/8.04	  ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by axiom 2 (condensed_detachment) }
7.86/8.04	  ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c))), implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))))), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by axiom 3 (ifeq_axiom) }
7.86/8.04	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c))), implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by lemma 6 }
7.86/8.04	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)), implies(implies(b, c), c)))), true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))))), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c))), implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by lemma 11 }
7.86/8.04	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))))), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c))), implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by axiom 3 (ifeq_axiom) }
7.86/8.04	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))))), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c))), implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by lemma 10 }
7.86/8.04	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by axiom 3 (ifeq_axiom) }
7.86/8.04	  ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by axiom 3 (ifeq_axiom) }
7.86/8.04	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.04	= { by lemma 10 }
7.86/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.05	= { by axiom 3 (ifeq_axiom) }
7.86/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.05	= { by axiom 2 (condensed_detachment) }
7.86/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.05	= { by axiom 3 (ifeq_axiom) }
7.86/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.05	= { by lemma 8 }
7.86/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(not(implies(implies(implies(not(b), b), c), c))), implies(implies(not(c), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.86/8.05	= { by axiom 3 (ifeq_axiom) }
7.89/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(not(implies(implies(implies(not(b), b), c), c))), implies(implies(not(c), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.05	= { by axiom 2 (condensed_detachment) }
7.89/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(not(c), c), c), implies(not(implies(implies(not(c), c), c)), not(not(implies(implies(implies(not(b), b), c), c)))))), true, ifeq(is_a_theorem(implies(implies(not(c), c), c)), true, is_a_theorem(implies(not(implies(implies(not(c), c), c)), not(not(implies(implies(implies(not(b), b), c), c))))), true), true), true, is_a_theorem(implies(implies(not(not(implies(implies(implies(not(b), b), c), c))), implies(implies(not(c), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.05	= { by axiom 4 (cn_3) }
7.89/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(c), c), c)), true, is_a_theorem(implies(not(implies(implies(not(c), c), c)), not(not(implies(implies(implies(not(b), b), c), c))))), true), true), true, is_a_theorem(implies(implies(not(not(implies(implies(implies(not(b), b), c), c))), implies(implies(not(c), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.05	= { by axiom 3 (ifeq_axiom) }
7.89/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(c), c), c)), true, is_a_theorem(implies(not(implies(implies(not(c), c), c)), not(not(implies(implies(implies(not(b), b), c), c))))), true), true, is_a_theorem(implies(implies(not(not(implies(implies(implies(not(b), b), c), c))), implies(implies(not(c), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.05	= { by axiom 1 (cn_2) }
7.89/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(not(c), c), c)), not(not(implies(implies(implies(not(b), b), c), c))))), true), true, is_a_theorem(implies(implies(not(not(implies(implies(implies(not(b), b), c), c))), implies(implies(not(c), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.05	= { by axiom 3 (ifeq_axiom) }
7.89/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(implies(not(c), c), c)), not(not(implies(implies(implies(not(b), b), c), c))))), true, is_a_theorem(implies(implies(not(not(implies(implies(implies(not(b), b), c), c))), implies(implies(not(c), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.05	= { by lemma 6 }
7.89/8.05	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.05	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)))), true, is_a_theorem(implies(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by lemma 6 }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(implies(implies(not(c), c), c)), implies(implies(not(c), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 1 (cn_2) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, ifeq(true, true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(not(c), c), c))), true, is_a_theorem(implies(implies(implies(implies(not(c), c), c), implies(implies(implies(not(b), b), c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by lemma 6 }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(b), b), c), implies(not(c), c)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(not(implies(implies(implies(not(b), b), c), c)), implies(implies(implies(not(b), b), c), c)))), true), true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by lemma 10 }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(implies(not(b), b), c), c), implies(implies(b, c), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true, is_a_theorem(implies(implies(implies(b, c), implies(implies(not(b), b), c)), implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by lemma 10 }
7.89/8.06	  ifeq(ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(b, c), implies(implies(not(b), b), c))), true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by lemma 11 }
7.89/8.06	  ifeq(ifeq(ifeq(true, true, ifeq(true, true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(true, true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by lemma 6 }
7.89/8.06	  ifeq(ifeq(ifeq(is_a_theorem(implies(b, implies(not(c), implies(not(b), b)))), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(b, implies(not(c), implies(not(b), b)))), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by lemma 6 }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(b, implies(implies(not(not(c)), b), implies(not(b), b)))), true, is_a_theorem(implies(implies(implies(implies(not(not(c)), b), implies(not(b), b)), implies(not(c), implies(not(b), b))), implies(b, implies(not(c), implies(not(b), b))))), true), true, is_a_theorem(implies(b, implies(not(c), implies(not(b), b)))), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(b, implies(implies(not(not(c)), b), implies(not(b), b)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(c)), b), implies(not(b), b)), implies(not(c), implies(not(b), b))), implies(b, implies(not(c), implies(not(b), b))))), true), true, is_a_theorem(implies(b, implies(not(c), implies(not(b), b)))), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 5 (cn_1) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(not(b), not(not(c))), implies(implies(not(not(c)), b), implies(not(b), b)))), true, is_a_theorem(implies(b, implies(implies(not(not(c)), b), implies(not(b), b)))), true), true, is_a_theorem(implies(implies(implies(implies(not(not(c)), b), implies(not(b), b)), implies(not(c), implies(not(b), b))), implies(b, implies(not(c), implies(not(b), b))))), true), true, is_a_theorem(implies(b, implies(not(c), implies(not(b), b)))), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by lemma 8 }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(implies(not(not(c)), b), implies(not(b), b)), implies(not(c), implies(not(b), b))), implies(b, implies(not(c), implies(not(b), b))))), true), true, is_a_theorem(implies(b, implies(not(c), implies(not(b), b)))), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(c)), b), implies(not(b), b)), implies(not(c), implies(not(b), b))), implies(b, implies(not(c), implies(not(b), b))))), true, is_a_theorem(implies(b, implies(not(c), implies(not(b), b)))), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(c)), b), implies(not(b), b)), implies(not(c), implies(not(b), b))), implies(b, implies(not(c), implies(not(b), b))))), true, ifeq(true, true, is_a_theorem(implies(b, implies(not(c), implies(not(b), b)))), true), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by lemma 7 }
7.89/8.06	  ifeq(ifeq(ifeq(ifeq(is_a_theorem(implies(implies(implies(implies(not(not(c)), b), implies(not(b), b)), implies(not(c), implies(not(b), b))), implies(b, implies(not(c), implies(not(b), b))))), true, ifeq(is_a_theorem(implies(implies(implies(not(not(c)), b), implies(not(b), b)), implies(not(c), implies(not(b), b)))), true, is_a_theorem(implies(b, implies(not(c), implies(not(b), b)))), true), true), true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 2 (condensed_detachment) }
7.89/8.06	  ifeq(ifeq(ifeq(true, true, is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  ifeq(ifeq(is_a_theorem(implies(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c)), implies(b, implies(implies(b, c), c)))), true, ifeq(is_a_theorem(implies(implies(not(c), implies(not(b), b)), implies(implies(b, c), c))), true, is_a_theorem(implies(b, implies(implies(b, c), c))), true), true), true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 2 (condensed_detachment) }
7.89/8.06	  ifeq(true, true, is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c)))), true)
7.89/8.06	= { by axiom 3 (ifeq_axiom) }
7.89/8.06	  is_a_theorem(implies(implies(a, implies(b, c)), implies(b, implies(a, c))))
7.89/8.06	% SZS output end Proof
7.89/8.06	
7.89/8.06	RESULT: Unsatisfiable (the axioms are contradictory).
7.89/8.06	EOF