Theorem rspceov 6692

Description: A frequently used special case of rspc2ev 3324 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

Step	Hyp	Ref	Expression
1		oveq1 6657	. . 3 |- ( x = C -> ( x F y ) = ( C F y ) )
2	1	eqeq2d 2632	. 2 |- ( x = C -> ( S = ( x F y ) <-> S = ( C F y ) ) )
3		oveq2 6658	. . 3 |- ( y = D -> ( C F y ) = ( C F D ) )
4	3	eqeq2d 2632	. 2 |- ( y = D -> ( S = ( C F y ) <-> S = ( C F D ) ) )
5	2, 4	rspc2ev 3324	1 |- ( ( C e. A /\ D e. B /\ S = ( C F D ) ) -> E. x e. A E. y e. B S = ( x F y ) )

Syntax hints:   -> wi 4   /\ w3a 1037   = wceq 1483   e. wcel 1990   E.wrex 2913  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  iunfictbso  8937  genpprecl  9823  elz2  11394  zaddcl  11417  znq  11792  qaddcl  11804  qmulcl  11806  qreccl  11808  xpsff1o  16228  mndpfo  17314  gafo  17729  lsmelvalix  18056  lsmelvalmi  18067  evthicc2  23229  i1fadd  23462  i1fmul  23463  isgrpoi  27352  shscli  28176  shsva  28179  shunssi  28227  pjpjhth  28284  spanunsni  28438  pjjsi  28559  ofrn2  29442  pstmfval  29939  ismblfin  33450  itg2addnc  33464  blbnd  33586  isgrpda  33754  sbgoldbalt  41669