Theorem rspceaov 32443

Description: A frequently used special case of rspc2ev 3221 for operation values, analogous to rspceov 6335. (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 2458	. . . 4 |- ( x = C -> F = F )
2		id 22	. . . 4 |- ( x = C -> x = C )
3		eqidd 2458	. . . 4 |- ( x = C -> y = y )
4	1, 2, 3	aoveq123d 32424	. . 3 |- ( x = C -> (( x F y)) = (( C F y)) )
5	4	eqeq2d 2471	. 2 |- ( x = C -> ( S = (( x F y)) <-> S = (( C F y)) ) )
6		eqidd 2458	. . . 4 |- ( y = D -> F = F )
7		eqidd 2458	. . . 4 |- ( y = D -> C = C )
8		id 22	. . . 4 |- ( y = D -> y = D )
9	6, 7, 8	aoveq123d 32424	. . 3 |- ( y = D -> (( C F y)) = (( C F D)) )
10	9	eqeq2d 2471	. 2 |- ( y = D -> ( S = (( C F y)) <-> S = (( C F D)) ) )
11	5, 10	rspc2ev 3221	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 973   = wceq 1395   e. wcel 1819   E.wrex 2808   ((caov 32361

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fv 5602  df-dfat 32362  df-afv 32363  df-aov 32364

This theorem is referenced by: (None)