Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  aoveq123d Unicode version

Theorem aoveq123d 27704
Description: Equality deduction for operation value, analogous to oveq123d 6034. (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 3927 . . 3  |-  ( ph  -> 
<. A ,  C >.  = 
<. B ,  D >. )
51, 4afveq12d 27659 . 2  |-  ( ph  ->  ( F''' <. A ,  C >. )  =  ( G''' <. B ,  D >. ) )
6 df-aov 27637 . 2  |- (( A F C))  =  ( F''' <. A ,  C >. )
7 df-aov 27637 . 2  |- (( B G D))  =  ( G''' <. B ,  D >. )
85, 6, 73eqtr4g 2437 1  |-  ( ph  -> (( A F C))  = (( B G D))  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   <.cop 3753  '''cafv 27633   ((caov 27634
This theorem is referenced by:  csbaovg  27706  rspceaov  27723  faovcl  27726
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-res 4823  df-iota 5351  df-fun 5389  df-fv 5395  df-dfat 27635  df-afv 27636  df-aov 27637
  Copyright terms: Public domain W3C validator