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Theorem aoveq123d 30252
Description: Equality deduction for operation value, analogous to oveq123d 6224. (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 4178 . . 3  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  D >. )
51, 4afveq12d 30207 . 2  |-  ( ph  ->  ( F''' <. A ,  C >. )  =  ( G''' <. B ,  D >. ) )
6 df-aov 30190 . 2  |- (( A F C))  =  ( F''' <. A ,  C >. )
7 df-aov 30190 . 2  |- (( B G D))  =  ( G''' <. B ,  D >. )
85, 6, 73eqtr4g 2520 1  |-  ( ph  -> (( A F C))  = (( B G D))  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370   <.cop 3994  '''cafv 30186   ((caov 30187
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-res 4963  df-iota 5492  df-fun 5531  df-fv 5537  df-dfat 30188  df-afv 30189  df-aov 30190
This theorem is referenced by:  csbaovg  30254  rspceaov  30271  faovcl  30274
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