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Theorem rspceaov 32443
Description: A frequently used special case of rspc2ev 3221 for operation values, analogous to rspceov 6335. (Contributed by Alexander van der Vekens, 26-May-2017.)
Assertion
Ref Expression
rspceaov  |-  ( ( 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 rspceaov
StepHypRef Expression
1 eqidd 2458 . . . 4  |-  ( x  =  C  ->  F  =  F )
2 id 22 . . . 4  |-  ( x  =  C  ->  x  =  C )
3 eqidd 2458 . . . 4  |-  ( x  =  C  ->  y  =  y )
41, 2, 3aoveq123d 32424 . . 3  |-  ( x  =  C  -> (( x F y))  = (( C F y))  )
54eqeq2d 2471 . 2  |-  ( x  =  C  ->  ( S  = (( x F
y)) 
<->  S  = (( C F y))  ) )
6 eqidd 2458 . . . 4  |-  ( y  =  D  ->  F  =  F )
7 eqidd 2458 . . . 4  |-  ( y  =  D  ->  C  =  C )
8 id 22 . . . 4  |-  ( y  =  D  ->  y  =  D )
96, 7, 8aoveq123d 32424 . . 3  |-  ( y  =  D  -> (( C F y))  = (( C F D))  )
109eqeq2d 2471 . 2  |-  ( y  =  D  ->  ( S  = (( C F
y)) 
<->  S  = (( C F D))  ) )
115, 10rspc2ev 3221 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 973    = wceq 1395    e. wcel 1819   E.wrex 2808   ((caov 32361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fv 5602  df-dfat 32362  df-afv 32363  df-aov 32364
This theorem is referenced by: (None)
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