Theorem rspceaov 38844

Description: A frequently used special case of rspc2ev 3149 for operation values, analogous to rspceov 6347. (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 2472	. . . 4 |- ( x = C -> F = F )
2		id 22	. . . 4 |- ( x = C -> x = C )
3		eqidd 2472	. . . 4 |- ( x = C -> y = y )
4	1, 2, 3	aoveq123d 38825	. . 3 |- ( x = C -> (( x F y)) = (( C F y)) )
5	4	eqeq2d 2481	. 2 |- ( x = C -> ( S = (( x F y)) <-> S = (( C F y)) ) )
6		eqidd 2472	. . . 4 |- ( y = D -> F = F )
7		eqidd 2472	. . . 4 |- ( y = D -> C = C )
8		id 22	. . . 4 |- ( y = D -> y = D )
9	6, 7, 8	aoveq123d 38825	. . 3 |- ( y = D -> (( C F y)) = (( C F D)) )
10	9	eqeq2d 2481	. 2 |- ( y = D -> ( S = (( C F y)) <-> S = (( C F D)) ) )
11	5, 10	rspc2ev 3149	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 1007   = wceq 1452   e. wcel 1904   E.wrex 2757   ((caov 38761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-iota 5553  df-fun 5591  df-fv 5597  df-dfat 38762  df-afv 38763  df-aov 38764

This theorem is referenced by: (None)