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Theorem rspceaov 38844
Description: A frequently used special case of rspc2ev 3149 for operation values, analogous to rspceov 6347. (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 2472 . . . 4  |-  ( x  =  C  ->  F  =  F )
2 id 22 . . . 4  |-  ( x  =  C  ->  x  =  C )
3 eqidd 2472 . . . 4  |-  ( x  =  C  ->  y  =  y )
41, 2, 3aoveq123d 38825 . . 3  |-  ( x  =  C  -> (( x F y))  = (( C F y))  )
54eqeq2d 2481 . 2  |-  ( x  =  C  ->  ( S  = (( x F
y)) 
<->  S  = (( C F y))  ) )
6 eqidd 2472 . . . 4  |-  ( y  =  D  ->  F  =  F )
7 eqidd 2472 . . . 4  |-  ( y  =  D  ->  C  =  C )
8 id 22 . . . 4  |-  ( y  =  D  ->  y  =  D )
96, 7, 8aoveq123d 38825 . . 3  |-  ( y  =  D  -> (( C F y))  = (( C F D))  )
109eqeq2d 2481 . 2  |-  ( y  =  D  ->  ( S  = (( C F
y)) 
<->  S  = (( C F D))  ) )
115, 10rspc2ev 3149 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 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   ((caov 38761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-res 4851  df-iota 5553  df-fun 5591  df-fv 5597  df-dfat 38762  df-afv 38763  df-aov 38764
This theorem is referenced by: (None)
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