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Theorem ax14 217
 Description: Axiom of Equality. Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). It is a special case of Axiom B8 (p. 75) of system S2 of [Tarski] p. 77. (Contributed by Mario Carneiro, 10-Oct-2014.)
Hypotheses
Ref Expression
ax14.1 A:(α → ∗)
ax14.2 B:(α → ∗)
ax14.3 C:α
Assertion
Ref Expression
ax14 ⊤⊧[[A = B] ⇒ [(AC) ⇒ (BC)]]

Proof of Theorem ax14
StepHypRef Expression
1 wtru 43 . . . . . 6 ⊤:∗
2 ax14.1 . . . . . . 7 A:(α → ∗)
3 ax14.2 . . . . . . 7 B:(α → ∗)
42, 3weqi 76 . . . . . 6 [A = B]:∗
51, 4wct 48 . . . . 5 (⊤, [A = B]):∗
6 ax14.3 . . . . . 6 C:α
72, 6wc 50 . . . . 5 (AC):∗
85, 7simpr 23 . . . 4 ((⊤, [A = B]), (AC))⊧(AC)
91, 4simpr 23 . . . . . 6 (⊤, [A = B])⊧[A = B]
102, 6, 9ceq1 89 . . . . 5 (⊤, [A = B])⊧[(AC) = (BC)]
1110, 7adantr 55 . . . 4 ((⊤, [A = B]), (AC))⊧[(AC) = (BC)]
128, 11mpbi 82 . . 3 ((⊤, [A = B]), (AC))⊧(BC)
1312ex 158 . 2 (⊤, [A = B])⊧[(AC) ⇒ (BC)]
1413ex 158 1 ⊤⊧[[A = B] ⇒ [(AC) ⇒ (BC)]]
 Colors of variables: type var term Syntax hints:   → ht 2  ∗hb 3  kc 5   = ke 7  ⊤kt 8  [kbr 9  kct 10  ⊧wffMMJ2 11  wffMMJ2t 12   ⇒ tim 121 This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-simpl 20  ax-simpr 21  ax-id 24  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-wctl 31  ax-wctr 32  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-ded 46  ax-wct 47  ax-wc 49  ax-ceq 51  ax-wv 63  ax-wl 65  ax-beta 67  ax-distrc 68  ax-leq 69  ax-distrl 70  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80  ax-hbl1 103  ax-17 105  ax-inst 113 This theorem depends on definitions:  df-ov 73  df-an 128  df-im 129 This theorem is referenced by: (None)
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