Theorem oveq123d 6106

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 6102	. 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 6103	. 2 |- ( ph -> ( A G C ) = ( B G D ) )
6	2, 5	eqtrd 2271	1 |- ( ph -> ( A F C ) = ( B G D ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402  (class class class)co 6085

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-un 3224  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-br 4131  df-iota 5337  df-fv 5385  df-ov 6088

This theorem is used by:  csbov123g  6124  issgrp  13771  sgrp1  13779  issgrpd  13780  ismndd  13803  grpsubfvalg  13903  grpsubpropdg  13962  imasgrp  13967  subgsub  14042  releqgg  14076  eqgex  14077  eqgfval  14078  prdsplusgfval  14268  prdsmulrfval  14270  isrng  14317  isrngd  14336  issrg  14353  srgidmlem  14366  isring  14388  ringass  14404  ringidmlem  14411  isringd  14430  ring1  14448  unitlinv  14517  unitrinv  14518  dvrfvald  14524  opprdrng  14704  islmodd  14713  islidlm  14900  rnglidlmsgrp  14918  rnglidlrng  14919  isassad  15095  asclfval  15105  ressascl  15123  psrval  15134