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Theorem aoveq123d 28038
Description: Equality deduction for operation value, analogous to oveq123d 5879. (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 3804 . . 3  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  D >. )
51, 4afveq12d 27996 . 2  |-  ( ph  ->  ( F''' <. A ,  C >. )  =  ( G''' <. B ,  D >. ) )
6 df-aov 27976 . 2  |- (( A F C))  =  ( F''' <. A ,  C >. )
7 df-aov 27976 . 2  |- (( B G D))  =  ( G''' <. B ,  D >. )
85, 6, 73eqtr4g 2340 1  |-  ( ph  -> (( A F C))  = (( B G D))  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623   <.cop 3643  '''cafv 27972   ((caov 27973
This theorem is referenced by:  csbaovg  28040  rspceaov  28057  faovcl  28060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-dfat 27974  df-afv 27975  df-aov 27976
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