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Theorem rspceov 6108
Description: A frequently used special case of rspc2ev 3052 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 6080 . . 3  |-  ( x  =  C  ->  (
x F y )  =  ( C F y ) )
21eqeq2d 2446 . 2  |-  ( x  =  C  ->  ( S  =  ( x F y )  <->  S  =  ( C F y ) ) )
3 oveq2 6081 . . 3  |-  ( y  =  D  ->  ( C F y )  =  ( C F D ) )
43eqeq2d 2446 . 2  |-  ( y  =  D  ->  ( S  =  ( C F y )  <->  S  =  ( C F D ) ) )
52, 4rspc2ev 3052 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 936    = wceq 1652    e. wcel 1725   E.wrex 2698  (class class class)co 6073
This theorem is referenced by:  iunfictbso  7987  genpprecl  8870  elz2  10290  zaddcl  10309  znq  10570  qaddcl  10582  qmulcl  10584  qreccl  10586  xpsff1o  13785  mndfo  14712  gafo  15065  lsmelvalix  15267  lsmelvalmi  15278  evthicc2  19349  i1fadd  19579  i1fmul  19580  isgrpoi  21778  isgrpda  21877  shscli  22811  shsva  22814  shunssi  22862  pjpjhth  22919  spanunsni  23073  pjjsi  23194  ofrn2  24045  ismblfin  26237  itg2addnc  26249  blbnd  26477
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076
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