Theorem oveq123d 6039

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 6035	. 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 6036	. 2 |- ( ph -> ( A G C ) = ( B G D ) )
6	2, 5	eqtrd 2264	1 |- ( ph -> ( A F C ) = ( B G D ) )

Syntax hints:   -> wi 4   = wceq 1397  (class class class)co 6018

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-iota 5286  df-fv 5334  df-ov 6021

This theorem is referenced by:  csbov123g  6057  prdsplusgfval  13385  prdsmulrfval  13387  issgrp  13504  sgrp1  13512  issgrpd  13513  ismndd  13538  grpsubfvalg  13646  grpsubpropdg  13705  imasgrp  13716  subgsub  13791  releqgg  13825  eqgex  13826  eqgfval  13827  isrng  13966  isrngd  13985  issrg  13997  srgidmlem  14010  isring  14032  ringass  14048  ringidmlem  14054  isringd  14073  ring1  14091  unitlinv  14159  unitrinv  14160  dvrfvald  14166  islmodd  14326  islidlm  14512  rnglidlmsgrp  14530  rnglidlrng  14531  psrval  14699