Theorem oveq123d 6106

Description: Equality deduction for operation value. (Contributed by FL, 22-Dec-2008.)

Hypotheses

Ref	Expression
oveq123d.1	|- ( ph -> F = G )
oveq123d.2	|- ( ph -> A = B )
oveq123d.3	|- ( ph -> C = D )

Assertion

Ref	Expression
oveq123d	|- ( ph -> ( A F C ) = ( B G D ) )

Proof of Theorem oveq123d

Step	Hyp	Ref	Expression
1		oveq123d.1	. . 3 |- ( ph -> F = G )
2	1	oveqd 6102	. 2 |- ( ph -> ( A F C ) = ( A G C ) )
3		oveq123d.2	. . 3 |- ( ph -> A = B )
4		oveq123d.3	. . 3 |- ( ph -> C = D )
5	3, 4	oveq12d 6103	. 2 |- ( ph -> ( A G C ) = ( B G D ) )
6	2, 5	eqtrd 2271	1 |- ( ph -> ( A F C ) = ( B G D ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  csbov123g  6124  issgrp  13767  sgrp1  13775  issgrpd  13776  ismndd  13799  grpsubfvalg  13899  grpsubpropdg  13958  imasgrp  13963  subgsub  14038  releqgg  14072  eqgex  14073  eqgfval  14074  prdsplusgfval  14233  prdsmulrfval  14235  isrng  14282  isrngd  14301  issrg  14318  srgidmlem  14331  isring  14353  ringass  14369  ringidmlem  14376  isringd  14395  ring1  14413  unitlinv  14482  unitrinv  14483  dvrfvald  14489  opprdrng  14669  islmodd  14678  islidlm  14865  rnglidlmsgrp  14883  rnglidlrng  14884  isassad  15060  asclfval  15070  ressascl  15088  psrval  15099