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Theorem rspceov 5893
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
StepHypRef Expression
1 oveq1 5865 . . 3  |-  ( x  =  C  ->  (
x F y )  =  ( C F y ) )
21eqeq2d 2294 . 2  |-  ( x  =  C  ->  ( S  =  ( x F y )  <->  S  =  ( C F y ) ) )
3 oveq2 5866 . . 3  |-  ( y  =  D  ->  ( C F y )  =  ( C F D ) )
43eqeq2d 2294 . 2  |-  ( y  =  D  ->  ( S  =  ( C F y )  <->  S  =  ( C F D ) ) )
52, 4rspc2ev 2892 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 934    = wceq 1623    e. wcel 1684   E.wrex 2544  (class class class)co 5858
This theorem is referenced by:  iunfictbso  7741  genpprecl  8625  elz2  10040  zaddcl  10059  znq  10320  qaddcl  10332  qmulcl  10334  qreccl  10336  xpsff1o  13470  mndfo  14397  gafo  14750  lsmelvalix  14952  lsmelvalmi  14963  evthicc2  18820  i1fadd  19050  i1fmul  19051  isgrpoi  20865  isgrpda  20964  shscli  21896  shsva  21899  shunssi  21947  pjpjhth  22004  spanunsni  22158  pjjsi  22279  grpodivfo  25374  blbnd  26511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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