Theorem ceq1 89

Description: Equality theorem for combination. (Contributed by Mario Carneiro, 7-Oct-2014.)

Hypotheses

Ref	Expression
ceq12.1	⊢ F:(α → β)
ceq12.2	⊢ A:α
ceq12.3	⊢ R⊧[F = T]

Assertion

Ref	Expression
ceq1	⊢ R⊧[(FA) = (TA)]

Proof of Theorem ceq1

Step	Hyp	Ref	Expression
1		ceq12.1	. 2 ⊢ F:(α → β)
2		ceq12.2	. 2 ⊢ A:α
3		ceq12.3	. 2 ⊢ R⊧[F = T]
4	3	ax-cb1 29	. . 3 ⊢ R:∗
5	4, 2	eqid 83	. 2 ⊢ R⊧[A = A]
6	1, 2, 3, 5	ceq12 88	1 ⊢ R⊧[(FA) = (TA)]

Syntax hints:   → ht 2  kc 5   = ke 7  [kbr 9  ⊧wffMMJ2 11  wffMMJ2t 12

This theorem was proved from axioms:  ax-syl 15  ax-jca 17  ax-trud 26  ax-cb1 29  ax-cb2 30  ax-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wov 71  ax-eqtypi 77

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  hbxfrf  107  ovl  117  alval  142  exval  143  euval  144  notval  145  ax4g  149  dfan2  154  eta  178  ac  197  ax14  217  axrep  220  axpow  221  axun  222