Theorem oveq123d 6028

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 6024	. 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 6025	. 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 6007

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 6010

This theorem is referenced by:  csbov123g  6046  prdsplusgfval  13333  prdsmulrfval  13335  issgrp  13452  sgrp1  13460  issgrpd  13461  ismndd  13486  grpsubfvalg  13594  grpsubpropdg  13653  imasgrp  13664  subgsub  13739  releqgg  13773  eqgex  13774  eqgfval  13775  isrng  13913  isrngd  13932  issrg  13944  srgidmlem  13957  isring  13979  ringass  13995  ringidmlem  14001  isringd  14020  ring1  14038  unitlinv  14106  unitrinv  14107  dvrfvald  14113  islmodd  14273  islidlm  14459  rnglidlmsgrp  14477  rnglidlrng  14478  psrval  14646