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Theorem rspceov 5578
Description: A frequently used special case of rspc2ev 2716 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
StepHypRef Expression
1 oveq1 5550 . . 3  |-  ( x  =  C  ->  (
x F y )  =  ( C F y ) )
21eqeq2d 2093 . 2  |-  ( x  =  C  ->  ( S  =  ( x F y )  <->  S  =  ( C F y ) ) )
3 oveq2 5551 . . 3  |-  ( y  =  D  ->  ( C F y )  =  ( C F D ) )
43eqeq2d 2093 . 2  |-  ( y  =  D  ->  ( S  =  ( C F y )  <->  S  =  ( C F D ) ) )
52, 4rspc2ev 2716 1  |-  ( ( C  e.  A  /\  D  e.  B  /\  S  =  ( C F D ) )  ->  E. x  e.  A  E. y  e.  B  S  =  ( x F y ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 920    = wceq 1285    e. wcel 1434   E.wrex 2350  (class class class)co 5543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-fv 4940  df-ov 5546
This theorem is referenced by:  genpprecll  6766  genppreclu  6767  elz2  8500  znq  8790  qaddcl  8801  qmulcl  8803  qreccl  8808
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