Theorem oveq123d 5967

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 5963	. 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 5964	. 2 |- ( ph -> ( A G C ) = ( B G D ) )
6	2, 5	eqtrd 2238	1 |- ( ph -> ( A F C ) = ( B G D ) )

Syntax hints:   -> wi 4   = wceq 1373  (class class class)co 5946

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-rex 2490  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-iota 5233  df-fv 5280  df-ov 5949

This theorem is referenced by:  csbov123g  5985  prdsplusgfval  13149  prdsmulrfval  13151  issgrp  13268  sgrp1  13276  issgrpd  13277  ismndd  13302  grpsubfvalg  13410  grpsubpropdg  13469  imasgrp  13480  subgsub  13555  releqgg  13589  eqgex  13590  eqgfval  13591  isrng  13729  isrngd  13748  issrg  13760  srgidmlem  13773  isring  13795  ringass  13811  ringidmlem  13817  isringd  13836  ring1  13854  unitlinv  13921  unitrinv  13922  dvrfvald  13928  islmodd  14088  islidlm  14274  rnglidlmsgrp  14292  rnglidlrng  14293  psrval  14461