Theorem rspceov 5964

Description: A frequently used special case of rspc2ev 2883 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 5929	. . 3 |- ( x = C -> ( x F y ) = ( C F y ) )
2	1	eqeq2d 2208	. 2 |- ( x = C -> ( S = ( x F y ) <-> S = ( C F y ) ) )
3		oveq2 5930	. . 3 |- ( y = D -> ( C F y ) = ( C F D ) )
4	3	eqeq2d 2208	. 2 |- ( y = D -> ( S = ( C F y ) <-> S = ( C F D ) ) )
5	2, 4	rspc2ev 2883	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 1364   e. wcel 2167   E.wrex 2476  (class class class)co 5922

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-br 4034  df-iota 5219  df-fv 5266  df-ov 5925

This theorem is referenced by:  genpprecll  7581  genppreclu  7582  elz2  9397  znq  9698  qaddcl  9709  qmulcl  9711  qreccl  9716  xpsff1o  12992  mndpfo  13079