Theorem oveq123d 5918

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 5914	. 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 5915	. 2 |- ( ph -> ( A G C ) = ( B G D ) )
6	2, 5	eqtrd 2222	1 |- ( ph -> ( A F C ) = ( B G D ) )

Syntax hints:   -> wi 4   = wceq 1364  (class class class)co 5897

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-iota 5196  df-fv 5243  df-ov 5900

This theorem is referenced by:  csbov123g  5935  issgrp  12881  sgrp1  12889  issgrpd  12890  ismndd  12913  grpsubfvalg  13004  grpsubpropdg  13063  imasgrp  13068  subgsub  13142  releqgg  13176  eqgex  13177  eqgfval  13178  isrng  13305  isrngd  13324  issrg  13336  srgidmlem  13349  isring  13371  ringass  13387  ringidmlem  13393  isringd  13412  ring1  13428  unitlinv  13493  unitrinv  13494  dvrfvald  13500  islmodd  13626  islidlm  13812  rnglidlmsgrp  13830  rnglidlrng  13831  psrval  13961