Theorem rspceov 5884

Description: A frequently used special case of rspc2ev 2845 for operation values. (Contributed by NM, 21-Mar-2007.)

Assertion

Ref	Expression
rspceov	|- ( ( 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 rspceov

Step	Hyp	Ref	Expression
1		oveq1 5849	. . 3 |- ( x = C -> ( x F y ) = ( C F y ) )
2	1	eqeq2d 2177	. 2 |- ( x = C -> ( S = ( x F y ) <-> S = ( C F y ) ) )
3		oveq2 5850	. . 3 |- ( y = D -> ( C F y ) = ( C F D ) )
4	3	eqeq2d 2177	. 2 |- ( y = D -> ( S = ( C F y ) <-> S = ( C F D ) ) )
5	2, 4	rspc2ev 2845	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 968   = wceq 1343   e. wcel 2136   E.wrex 2445  (class class class)co 5842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by:  genpprecll  7455  genppreclu  7456  elz2  9262  znq  9562  qaddcl  9573  qmulcl  9575  qreccl  9580