Theorem rspceov 5986

Description: A frequently used special case of rspc2ev 2891 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 5950	. . 3 |- ( x = C -> ( x F y ) = ( C F y ) )
2	1	eqeq2d 2216	. 2 |- ( x = C -> ( S = ( x F y ) <-> S = ( C F y ) ) )
3		oveq2 5951	. . 3 |- ( y = D -> ( C F y ) = ( C F D ) )
4	3	eqeq2d 2216	. 2 |- ( y = D -> ( S = ( C F y ) <-> S = ( C F D ) ) )
5	2, 4	rspc2ev 2891	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 980   = wceq 1372   e. wcel 2175   E.wrex 2484  (class class class)co 5943

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-br 4044  df-iota 5231  df-fv 5278  df-ov 5946

This theorem is referenced by:  genpprecll  7626  genppreclu  7627  elz2  9443  znq  9744  qaddcl  9755  qmulcl  9757  qreccl  9762  xpsff1o  13152  mndpfo  13241