Theorem rspceaov 38410

Description: A frequently used special case of rspc2ev 3193 for operation values, analogous to rspceov 6340. (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 2423	. . . 4 |- ( x = C -> F = F )
2		id 23	. . . 4 |- ( x = C -> x = C )
3		eqidd 2423	. . . 4 |- ( x = C -> y = y )
4	1, 2, 3	aoveq123d 38391	. . 3 |- ( x = C -> (( x F y)) = (( C F y)) )
5	4	eqeq2d 2436	. 2 |- ( x = C -> ( S = (( x F y)) <-> S = (( C F y)) ) )
6		eqidd 2423	. . . 4 |- ( y = D -> F = F )
7		eqidd 2423	. . . 4 |- ( y = D -> C = C )
8		id 23	. . . 4 |- ( y = D -> y = D )
9	6, 7, 8	aoveq123d 38391	. . 3 |- ( y = D -> (( C F y)) = (( C F D)) )
10	9	eqeq2d 2436	. 2 |- ( y = D -> ( S = (( C F y)) <-> S = (( C F D)) ) )
11	5, 10	rspc2ev 3193	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 982   = wceq 1437   e. wcel 1868   E.wrex 2776   ((caov 38328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-res 4861  df-iota 5561  df-fun 5599  df-fv 5605  df-dfat 38329  df-afv 38330  df-aov 38331

This theorem is referenced by: (None)