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Theorem rspceaov 38699
Description: A frequently used special case of rspc2ev 3161 for operation values, analogous to rspceov 6329. (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 2452 . . . 4  |-  ( x  =  C  ->  F  =  F )
2 id 22 . . . 4  |-  ( x  =  C  ->  x  =  C )
3 eqidd 2452 . . . 4  |-  ( x  =  C  ->  y  =  y )
41, 2, 3aoveq123d 38680 . . 3  |-  ( x  =  C  -> (( x F y))  = (( C F y))  )
54eqeq2d 2461 . 2  |-  ( x  =  C  ->  ( S  = (( x F
y)) 
<->  S  = (( C F y))  ) )
6 eqidd 2452 . . . 4  |-  ( y  =  D  ->  F  =  F )
7 eqidd 2452 . . . 4  |-  ( y  =  D  ->  C  =  C )
8 id 22 . . . 4  |-  ( y  =  D  ->  y  =  D )
96, 7, 8aoveq123d 38680 . . 3  |-  ( y  =  D  -> (( C F y))  = (( C F D))  )
109eqeq2d 2461 . 2  |-  ( y  =  D  ->  ( S  = (( C F
y)) 
<->  S  = (( C F D))  ) )
115, 10rspc2ev 3161 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 985    = wceq 1444    e. wcel 1887   E.wrex 2738   ((caov 38616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-rex 2743  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-res 4846  df-iota 5546  df-fun 5584  df-fv 5590  df-dfat 38617  df-afv 38618  df-aov 38619
This theorem is referenced by: (None)
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