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Theorem rspceaov 30108
Description: A frequently used special case of rspc2ev 3086 for operation values, analogous to rspceov 6133. (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 2444 . . . 4  |-  ( x  =  C  ->  F  =  F )
2 id 22 . . . 4  |-  ( x  =  C  ->  x  =  C )
3 eqidd 2444 . . . 4  |-  ( x  =  C  ->  y  =  y )
41, 2, 3aoveq123d 30089 . . 3  |-  ( x  =  C  -> (( x F y))  = (( C F y))  )
54eqeq2d 2454 . 2  |-  ( x  =  C  ->  ( S  = (( x F
y)) 
<->  S  = (( C F y))  ) )
6 eqidd 2444 . . . 4  |-  ( y  =  D  ->  F  =  F )
7 eqidd 2444 . . . 4  |-  ( y  =  D  ->  C  =  C )
8 id 22 . . . 4  |-  ( y  =  D  ->  y  =  D )
96, 7, 8aoveq123d 30089 . . 3  |-  ( y  =  D  -> (( C F y))  = (( C F D))  )
109eqeq2d 2454 . 2  |-  ( y  =  D  ->  ( S  = (( C F
y)) 
<->  S  = (( C F D))  ) )
115, 10rspc2ev 3086 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 2721   ((caov 30024
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-res 4857  df-iota 5386  df-fun 5425  df-fv 5431  df-dfat 30025  df-afv 30026  df-aov 30027
This theorem is referenced by: (None)
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