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Theorem aoveq123d 28018
Description: Equality deduction for operation value, analogous to oveq123d 6102. (Contributed by Alexander van der Vekens, 26-May-2017.)
Hypotheses
Ref Expression
aoveq123d.1  |-  ( ph  ->  F  =  G )
aoveq123d.2  |-  ( ph  ->  A  =  B )
aoveq123d.3  |-  ( ph  ->  C  =  D )
Assertion
Ref Expression
aoveq123d  |-  ( ph  -> (( A F C))  = (( B G D))  )

Proof of Theorem aoveq123d
StepHypRef Expression
1 aoveq123d.1 . . 3  |-  ( ph  ->  F  =  G )
2 aoveq123d.2 . . . 4  |-  ( ph  ->  A  =  B )
3 aoveq123d.3 . . . 4  |-  ( ph  ->  C  =  D )
42, 3opeq12d 3992 . . 3  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  D >. )
51, 4afveq12d 27973 . 2  |-  ( ph  ->  ( F''' <. A ,  C >. )  =  ( G''' <. B ,  D >. ) )
6 df-aov 27952 . 2  |- (( A F C))  =  ( F''' <. A ,  C >. )
7 df-aov 27952 . 2  |- (( B G D))  =  ( G''' <. B ,  D >. )
85, 6, 73eqtr4g 2493 1  |-  ( ph  -> (( A F C))  = (( B G D))  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652   <.cop 3817  '''cafv 27948   ((caov 27949
This theorem is referenced by:  csbaovg  28020  rspceaov  28037  faovcl  28040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-res 4890  df-iota 5418  df-fun 5456  df-fv 5462  df-dfat 27950  df-afv 27951  df-aov 27952
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