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

Step	Hyp	Ref	Expression
1		wtru 43	. . . . . 6 ⊢ ⊤:∗
2		ax14.1	. . . . . . 7 ⊢ A:(α → ∗)
3		ax14.2	. . . . . . 7 ⊢ B:(α → ∗)
4	2, 3	weqi 76	. . . . . 6 ⊢ [A = B]:∗
5	1, 4	wct 48	. . . . 5 ⊢ (⊤, [A = B]):∗
6		ax14.3	. . . . . 6 ⊢ C:α
7	2, 6	wc 50	. . . . 5 ⊢ (AC):∗
8	5, 7	simpr 23	. . . 4 ⊢ ((⊤, [A = B]), (AC))⊧(AC)
9	1, 4	simpr 23	. . . . . 6 ⊢ (⊤, [A = B])⊧[A = B]
10	2, 6, 9	ceq1 89	. . . . 5 ⊢ (⊤, [A = B])⊧[(AC) = (BC)]
11	10, 7	adantr 55	. . . 4 ⊢ ((⊤, [A = B]), (AC))⊧[(AC) = (BC)]
12	8, 11	mpbi 82	. . 3 ⊢ ((⊤, [A = B]), (AC))⊧(BC)
13	12	ex 158	. 2 ⊢ (⊤, [A = B])⊧[(AC) ⇒ (BC)]
14	13	ex 158	1 ⊢ ⊤⊧[[A = B] ⇒ [(AC) ⇒ (BC)]]

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)