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Theorem ax8 211
 Description: Axiom of Equality. Axiom scheme C8' in [Megill] p. 448 (p. 16 of the preprint). Also appears as Axiom C7 of [Monk2] p. 105. (Contributed by Mario Carneiro, 10-Oct-2014.)
Hypotheses
Ref Expression
ax8.1 A:α
ax8.2 B:α
ax8.3 C:α
Assertion
Ref Expression
ax8 ⊤⊧[[A = B] ⇒ [[A = C] ⇒ [B = C]]]

Proof of Theorem ax8
StepHypRef Expression
1 ax8.2 . . . . 5 B:α
2 ax8.1 . . . . . 6 A:α
32, 1weqi 76 . . . . . . 7 [A = B]:∗
4 ax8.3 . . . . . . . 8 C:α
52, 4weqi 76 . . . . . . 7 [A = C]:∗
63, 5simpl 22 . . . . . 6 ([A = B], [A = C])⊧[A = B]
72, 6eqcomi 79 . . . . 5 ([A = B], [A = C])⊧[B = A]
83, 5simpr 23 . . . . 5 ([A = B], [A = C])⊧[A = C]
91, 7, 8eqtri 95 . . . 4 ([A = B], [A = C])⊧[B = C]
109ex 158 . . 3 [A = B]⊧[[A = C] ⇒ [B = C]]
11 wtru 43 . . 3 ⊤:∗
1210, 11adantl 56 . 2 (⊤, [A = B])⊧[[A = C] ⇒ [B = C]]
1312ex 158 1 ⊤⊧[[A = B] ⇒ [[A = C] ⇒ [B = C]]]
 Colors of variables: type var term Syntax hints:   = 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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