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Theorem rspceov 5909
Description: A frequently used special case of rspc2ev 2905 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 5881 . . 3  |-  ( x  =  C  ->  (
x F y )  =  ( C F y ) )
21eqeq2d 2307 . 2  |-  ( x  =  C  ->  ( S  =  ( x F y )  <->  S  =  ( C F y ) ) )
3 oveq2 5882 . . 3  |-  ( y  =  D  ->  ( C F y )  =  ( C F D ) )
43eqeq2d 2307 . 2  |-  ( y  =  D  ->  ( S  =  ( C F y )  <->  S  =  ( C F D ) ) )
52, 4rspc2ev 2905 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 1632    e. wcel 1696   E.wrex 2557  (class class class)co 5874
This theorem is referenced by:  iunfictbso  7757  genpprecl  8641  elz2  10056  zaddcl  10075  znq  10336  qaddcl  10348  qmulcl  10350  qreccl  10352  xpsff1o  13486  mndfo  14413  gafo  14766  lsmelvalix  14968  lsmelvalmi  14979  evthicc2  18836  i1fadd  19066  i1fmul  19067  isgrpoi  20881  isgrpda  20980  shscli  21912  shsva  21915  shunssi  21963  pjpjhth  22020  spanunsni  22174  pjjsi  22295  grpodivfo  25477  blbnd  26614
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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