ax8 - Higher-Order Logic Explorer

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Theorem ax8 198

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.

**Assertion**
Ref	Expression
ax8	⊢ ⊤⊧[[A = B] ⇒ [[A = C] ⇒ [B = C]]]

**Proof of Theorem ax8**
Step	Hyp	Ref	Expression
1		ax8.2	. . . . 5 ⊢ B:α
2		ax8.1	. . . . . 6 ⊢ A:α
3	2, 1	weqi 68	. . . . . . 7 ⊢ [A = B]:∗
4		ax8.3	. . . . . . . 8 ⊢ C:α
5	2, 4	weqi 68	. . . . . . 7 ⊢ [A = C]:∗
6	3, 5	simpl 22	. . . . . 6 ⊢ ([A = B], [A = C])⊧[A = B]
7	2, 6	eqcomi 70	. . . . 5 ⊢ ([A = B], [A = C])⊧[B = A]
8	3, 5	simpr 23	. . . . 5 ⊢ ([A = B], [A = C])⊧[A = C]
9	1, 7, 8	eqtri 85	. . . 4 ⊢ ([A = B], [A = C])⊧[B = C]
10	9	ex 148	. . 3 ⊢ [A = B]⊧[[A = C] ⇒ [B = C]]
11		wtru 40	. . 3 ⊢ ⊤:∗
12	10, 11	adantl 51	. 2 ⊢ (⊤, [A = B])⊧[[A = C] ⇒ [B = C]]
13	12	ex 148	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 111

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-refl 39 ax-eqmp 42 ax-ded 43 ax-ceq 46 ax-beta 60 ax-distrc 61 ax-leq 62 ax-distrl 63 ax-hbl1 93 ax-17 95 ax-inst 103

This theorem depends on definitions: df-ov 65 df-an 118 df-im 119

This theorem is referenced by: (None)