Theorem rspceaov 38699

Description: A frequently used special case of rspc2ev 3161 for operation values, analogous to rspceov 6329. (Contributed by Alexander van der Vekens, 26-May-2017.)

Assertion

Ref	Expression
rspceaov	|- ( ( C e. A /\ D e. B /\ S = (( C F D)) ) -> E. x e. A E. y e. B S = (( x F y)) )

Distinct variable groups:   x, A   x, y, B   x, C, y   y, D   x, F, y   x, S, y

Allowed substitution hints:   A( y)   D( x)

Proof of Theorem rspceaov

Step	Hyp	Ref	Expression
1		eqidd 2452	. . . 4 |- ( x = C -> F = F )
2		id 22	. . . 4 |- ( x = C -> x = C )
3		eqidd 2452	. . . 4 |- ( x = C -> y = y )
4	1, 2, 3	aoveq123d 38680	. . 3 |- ( x = C -> (( x F y)) = (( C F y)) )
5	4	eqeq2d 2461	. 2 |- ( x = C -> ( S = (( x F y)) <-> S = (( C F y)) ) )
6		eqidd 2452	. . . 4 |- ( y = D -> F = F )
7		eqidd 2452	. . . 4 |- ( y = D -> C = C )
8		id 22	. . . 4 |- ( y = D -> y = D )
9	6, 7, 8	aoveq123d 38680	. . 3 |- ( y = D -> (( C F y)) = (( C F D)) )
10	9	eqeq2d 2461	. 2 |- ( y = D -> ( S = (( C F y)) <-> S = (( C F D)) ) )
11	5, 10	rspc2ev 3161	1 |- ( ( C e. A /\ D e. B /\ S = (( C F D)) ) -> E. x e. A E. y e. B S = (( x F y)) )

Syntax hints:   -> wi 4   /\ w3a 985   = wceq 1444   e. wcel 1887   E.wrex 2738   ((caov 38616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-res 4846  df-iota 5546  df-fun 5584  df-fv 5590  df-dfat 38617  df-afv 38618  df-aov 38619

This theorem is referenced by: (None)