Theorem rspceaov 30108

Description: A frequently used special case of rspc2ev 3086 for operation values, analogous to rspceov 6133. (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 2444	. . . 4 |- ( x = C -> F = F )
2		id 22	. . . 4 |- ( x = C -> x = C )
3		eqidd 2444	. . . 4 |- ( x = C -> y = y )
4	1, 2, 3	aoveq123d 30089	. . 3 |- ( x = C -> (( x F y)) = (( C F y)) )
5	4	eqeq2d 2454	. 2 |- ( x = C -> ( S = (( x F y)) <-> S = (( C F y)) ) )
6		eqidd 2444	. . . 4 |- ( y = D -> F = F )
7		eqidd 2444	. . . 4 |- ( y = D -> C = C )
8		id 22	. . . 4 |- ( y = D -> y = D )
9	6, 7, 8	aoveq123d 30089	. . 3 |- ( y = D -> (( C F y)) = (( C F D)) )
10	9	eqeq2d 2454	. 2 |- ( y = D -> ( S = (( C F y)) <-> S = (( C F D)) ) )
11	5, 10	rspc2ev 3086	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 965   = wceq 1369   e. wcel 1756   E.wrex 2721   ((caov 30024

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5386  df-fun 5425  df-fv 5431  df-dfat 30025  df-afv 30026  df-aov 30027

This theorem is referenced by: (None)