Theorem oveq123d 5988

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 5984	. 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 5985	. 2 |- ( ph -> ( A G C ) = ( B G D ) )
6	2, 5	eqtrd 2240	1 |- ( ph -> ( A F C ) = ( B G D ) )

Syntax hints:   -> wi 4   = wceq 1373  (class class class)co 5967

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-rex 2492  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-iota 5251  df-fv 5298  df-ov 5970

This theorem is referenced by:  csbov123g  6006  prdsplusgfval  13231  prdsmulrfval  13233  issgrp  13350  sgrp1  13358  issgrpd  13359  ismndd  13384  grpsubfvalg  13492  grpsubpropdg  13551  imasgrp  13562  subgsub  13637  releqgg  13671  eqgex  13672  eqgfval  13673  isrng  13811  isrngd  13830  issrg  13842  srgidmlem  13855  isring  13877  ringass  13893  ringidmlem  13899  isringd  13918  ring1  13936  unitlinv  14003  unitrinv  14004  dvrfvald  14010  islmodd  14170  islidlm  14356  rnglidlmsgrp  14374  rnglidlrng  14375  psrval  14543