Theorem rspceov 5987

Description: A frequently used special case of rspc2ev 2892 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 5951	. . 3 |- ( x = C -> ( x F y ) = ( C F y ) )
2	1	eqeq2d 2217	. 2 |- ( x = C -> ( S = ( x F y ) <-> S = ( C F y ) ) )
3		oveq2 5952	. . 3 |- ( y = D -> ( C F y ) = ( C F D ) )
4	3	eqeq2d 2217	. 2 |- ( y = D -> ( S = ( C F y ) <-> S = ( C F D ) ) )
5	2, 4	rspc2ev 2892	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 2176   E.wrex 2485  (class class class)co 5944

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-fv 5279  df-ov 5947

This theorem is referenced by:  genpprecll  7627  genppreclu  7628  elz2  9444  znq  9745  qaddcl  9756  qmulcl  9758  qreccl  9763  xpsff1o  13181  mndpfo  13270