Theorem oveq123d 6079

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 6075	. 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 6076	. 2 |- ( ph -> ( A G C ) = ( B G D ) )
6	2, 5	eqtrd 2267	1 |- ( ph -> ( A F C ) = ( B G D ) )

Syntax hints:   -> wi 4   = wceq 1398  (class class class)co 6058

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365  df-ov 6061

This theorem is referenced by:  csbov123g  6097  issgrp  13666  sgrp1  13674  issgrpd  13675  ismndd  13698  grpsubfvalg  13800  grpsubpropdg  13859  imasgrp  13864  subgsub  13939  releqgg  13973  eqgex  13974  eqgfval  13975  prdsplusgfval  14126  prdsmulrfval  14128  isrng  14173  isrngd  14192  issrg  14208  srgidmlem  14221  isring  14243  ringass  14259  ringidmlem  14265  isringd  14284  ring1  14302  unitlinv  14371  unitrinv  14372  dvrfvald  14378  opprdrng  14558  islmodd  14567  islidlm  14753  rnglidlmsgrp  14771  rnglidlrng  14772  psrval  14940