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Theorem rspceaov 38410
Description: A frequently used special case of rspc2ev 3193 for operation values, analogous to rspceov 6340. (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 2423 . . . 4  |-  ( x  =  C  ->  F  =  F )
2 id 23 . . . 4  |-  ( x  =  C  ->  x  =  C )
3 eqidd 2423 . . . 4  |-  ( x  =  C  ->  y  =  y )
41, 2, 3aoveq123d 38391 . . 3  |-  ( x  =  C  -> (( x F y))  = (( C F y))  )
54eqeq2d 2436 . 2  |-  ( x  =  C  ->  ( S  = (( x F
y)) 
<->  S  = (( C F y))  ) )
6 eqidd 2423 . . . 4  |-  ( y  =  D  ->  F  =  F )
7 eqidd 2423 . . . 4  |-  ( y  =  D  ->  C  =  C )
8 id 23 . . . 4  |-  ( y  =  D  ->  y  =  D )
96, 7, 8aoveq123d 38391 . . 3  |-  ( y  =  D  -> (( C F y))  = (( C F D))  )
109eqeq2d 2436 . 2  |-  ( y  =  D  ->  ( S  = (( C F
y)) 
<->  S  = (( C F D))  ) )
115, 10rspc2ev 3193 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 982    = wceq 1437    e. wcel 1868   E.wrex 2776   ((caov 38328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rex 2781  df-rab 2784  df-v 3083  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-res 4861  df-iota 5561  df-fun 5599  df-fv 5605  df-dfat 38329  df-afv 38330  df-aov 38331
This theorem is referenced by: (None)
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