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Theorem rspceov 6127
Description: A frequently used special case of rspc2ev 3080 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 6097 . . 3  |-  ( x  =  C  ->  (
x F y )  =  ( C F y ) )
21eqeq2d 2453 . 2  |-  ( x  =  C  ->  ( S  =  ( x F y )  <->  S  =  ( C F y ) ) )
3 oveq2 6098 . . 3  |-  ( y  =  D  ->  ( C F y )  =  ( C F D ) )
43eqeq2d 2453 . 2  |-  ( y  =  D  ->  ( S  =  ( C F y )  <->  S  =  ( C F D ) ) )
52, 4rspc2ev 3080 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 setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2715  (class class class)co 6090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-rex 2720  df-rab 2723  df-v 2973  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-br 4292  df-iota 5380  df-fv 5425  df-ov 6093
This theorem is referenced by:  iunfictbso  8283  genpprecl  9169  elz2  10662  zaddcl  10684  znq  10956  qaddcl  10968  qmulcl  10970  qreccl  10972  xpsff1o  14505  mndfo  15444  gafo  15813  lsmelvalix  16139  lsmelvalmi  16150  evthicc2  20943  i1fadd  21172  i1fmul  21173  isgrpoi  23684  isgrpda  23783  shscli  24719  shsva  24722  shunssi  24770  pjpjhth  24827  spanunsni  24981  pjjsi  25102  ofrn2  25957  ismblfin  28430  itg2addnc  28444  blbnd  28684
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