Theorem aoveq123d 41258

Description: Equality deduction for operation value, analogous to oveq123d 6671. (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

Step	Hyp	Ref	Expression
1		aoveq123d.1	. . 3 |- ( ph -> F = G )
2		aoveq123d.2	. . . 4 |- ( ph -> A = B )
3		aoveq123d.3	. . . 4 |- ( ph -> C = D )
4	2, 3	opeq12d 4410	. . 3 |- ( ph -> <. A , C >. = <. B , D >. )
5	1, 4	afveq12d 41213	. 2 |- ( ph -> ( F''' <. A , C >. ) = ( G''' <. B , D >. ) )
6		df-aov 41198	. 2 |- (( A F C)) = ( F''' <. A , C >. )
7		df-aov 41198	. 2 |- (( B G D)) = ( G''' <. B , D >. )
8	5, 6, 7	3eqtr4g 2681	1 |- ( ph -> (( A F C)) = (( B G D)) )

Syntax hints:   -> wi 4   = wceq 1483   <.cop 4183  '''cafv 41194   ((caov 41195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-dfat 41196  df-afv 41197  df-aov 41198

This theorem is referenced by:  csbaovg  41260  rspceaov  41277  faovcl  41280