Theorem oveq123d 6022

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 6018	. 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 6019	. 2 |- ( ph -> ( A G C ) = ( B G D ) )
6	2, 5	eqtrd 2262	1 |- ( ph -> ( A F C ) = ( B G D ) )

Syntax hints:   -> wi 4   = wceq 1395  (class class class)co 6001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6004

This theorem is referenced by:  csbov123g  6040  prdsplusgfval  13317  prdsmulrfval  13319  issgrp  13436  sgrp1  13444  issgrpd  13445  ismndd  13470  grpsubfvalg  13578  grpsubpropdg  13637  imasgrp  13648  subgsub  13723  releqgg  13757  eqgex  13758  eqgfval  13759  isrng  13897  isrngd  13916  issrg  13928  srgidmlem  13941  isring  13963  ringass  13979  ringidmlem  13985  isringd  14004  ring1  14022  unitlinv  14090  unitrinv  14091  dvrfvald  14097  islmodd  14257  islidlm  14443  rnglidlmsgrp  14461  rnglidlrng  14462  psrval  14630