Theorem oveq123d 6073

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 6069	. 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 6070	. 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 6052

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 3217  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-iota 5314  df-fv 5362  df-ov 6055

This theorem is referenced by:  csbov123g  6091  prdsplusgfval  13514  prdsmulrfval  13516  issgrp  13633  sgrp1  13641  issgrpd  13642  ismndd  13667  grpsubfvalg  13775  grpsubpropdg  13834  imasgrp  13845  subgsub  13920  releqgg  13954  eqgex  13955  eqgfval  13956  isrng  14095  isrngd  14114  issrg  14126  srgidmlem  14139  isring  14161  ringass  14177  ringidmlem  14183  isringd  14202  ring1  14220  unitlinv  14288  unitrinv  14289  dvrfvald  14295  islmodd  14458  islidlm  14644  rnglidlmsgrp  14662  rnglidlrng  14663  psrval  14831