Theorem rspceov 6010

Description: A frequently used special case of rspc2ev 2899 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 5974	. . 3 |- ( x = C -> ( x F y ) = ( C F y ) )
2	1	eqeq2d 2219	. 2 |- ( x = C -> ( S = ( x F y ) <-> S = ( C F y ) ) )
3		oveq2 5975	. . 3 |- ( y = D -> ( C F y ) = ( C F D ) )
4	3	eqeq2d 2219	. 2 |- ( y = D -> ( S = ( C F y ) <-> S = ( C F D ) ) )
5	2, 4	rspc2ev 2899	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 981   = wceq 1373   e. wcel 2178   E.wrex 2487  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  genpprecll  7662  genppreclu  7663  elz2  9479  znq  9780  qaddcl  9791  qmulcl  9793  qreccl  9798  xpsff1o  13296  mndpfo  13385