Theorem oveq1 89

Description: Equality theorem for binary operation.

Hypotheses

Ref	Expression
oveq.1	⊢ F:(α → (β → γ))
oveq.2	⊢ A:α
oveq.3	⊢ B:β
oveq1.4	⊢ R⊧[A = C]

Assertion

Ref	Expression
oveq1	⊢ R⊧[[AFB] = [CFB]]

Proof of Theorem oveq1

Step	Hyp	Ref	Expression
1		oveq.1	. 2 ⊢ F:(α → (β → γ))
2		oveq.2	. 2 ⊢ A:α
3		oveq.3	. 2 ⊢ B:β
4		oveq1.4	. . . 4 ⊢ R⊧[A = C]
5	4	ax-cb1 29	. . 3 ⊢ R:∗
6	5, 1	eqid 73	. 2 ⊢ R⊧[F = F]
7	5, 3	eqid 73	. 2 ⊢ R⊧[B = B]
8	1, 2, 3, 6, 4, 7	oveq123 88	1 ⊢ R⊧[[AFB] = [CFB]]

Syntax hints:   → ht 2   = 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-refl 39  ax-eqmp 42  ax-ceq 46

This theorem depends on definitions:  df-ov 65

This theorem is referenced by:  alval  132  exval  133  euval  134  notval  135  imval  136  orval  137  anval  138  exlimdv  157  ax4e  158  exlimd  171  ac  184  exmid  186  ax10  200  axrep  207