Theorem oveq 102

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

Hypotheses

Ref	Expression
oveq.1	|- F:(al -> (be -> ga))
oveq.2	|- A:al
oveq.3	|- B:be
oveq.4	|- R |= [F = S]

Assertion

Ref	Expression
oveq	|- R |= [[AFB] = [ASB]]

Proof of Theorem oveq

Step	Hyp	Ref	Expression
1		oveq.1	. 2 |- F:(al -> (be -> ga))
2		oveq.2	. 2 |- A:al
3		oveq.3	. 2 |- B:be
4		oveq.4	. 2 |- R |= [F = S]
5	4	ax-cb1 29	. . 3 |- R:*
6	5, 2	eqid 83	. 2 |- R |= [A = A]
7	5, 3	eqid 83	. 2 |- R |= [B = B]
8	1, 2, 3, 4, 6, 7	oveq123 98	1 |- R |= [[AFB] = [ASB]]

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-weq 40  ax-refl 42  ax-eqmp 45  ax-wc 49  ax-ceq 51  ax-wov 71  ax-eqtypi 77  ax-eqtypri 80

This theorem depends on definitions:  df-ov 73

This theorem is referenced by:  imval  146  orval  147  anval  148  dfan2  154