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Theorem aoveq123d 38550
Description: Equality deduction for operation value, analogous to oveq123d 6326. (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 4195 . . 3  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  D >. )
51, 4afveq12d 38505 . 2  |-  ( ph  ->  ( F''' <. A ,  C >. )  =  ( G''' <. B ,  D >. ) )
6 df-aov 38490 . 2  |- (( A F C))  =  ( F''' <. A ,  C >. )
7 df-aov 38490 . 2  |- (( B G D))  =  ( G''' <. B ,  D >. )
85, 6, 73eqtr4g 2488 1  |-  ( ph  -> (( A F C))  = (( B G D))  )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437   <.cop 4004  '''cafv 38486   ((caov 38487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-rex 2777  df-rab 2780  df-v 3082  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-res 4865  df-iota 5565  df-fun 5603  df-fv 5609  df-dfat 38488  df-afv 38489  df-aov 38490
This theorem is referenced by:  csbaovg  38552  rspceaov  38569  faovcl  38572
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