Theorem rspceaov 41277

Description: A frequently used special case of rspc2ev 3324 for operation values, analogous to rspceov 6692. (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 2623	. . . 4 |- ( x = C -> F = F )
2		id 22	. . . 4 |- ( x = C -> x = C )
3		eqidd 2623	. . . 4 |- ( x = C -> y = y )
4	1, 2, 3	aoveq123d 41258	. . 3 |- ( x = C -> (( x F y)) = (( C F y)) )
5	4	eqeq2d 2632	. 2 |- ( x = C -> ( S = (( x F y)) <-> S = (( C F y)) ) )
6		eqidd 2623	. . . 4 |- ( y = D -> F = F )
7		eqidd 2623	. . . 4 |- ( y = D -> C = C )
8		id 22	. . . 4 |- ( y = D -> y = D )
9	6, 7, 8	aoveq123d 41258	. . 3 |- ( y = D -> (( C F y)) = (( C F D)) )
10	9	eqeq2d 2632	. 2 |- ( y = D -> ( S = (( C F y)) <-> S = (( C F D)) ) )
11	5, 10	rspc2ev 3324	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 1037   = wceq 1483   e. wcel 1990   E.wrex 2913   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by: (None)