Theorem oveq123d 5946

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 5942	. 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 5943	. 2 |- ( ph -> ( A G C ) = ( B G D ) )
6	2, 5	eqtrd 2229	1 |- ( ph -> ( A F C ) = ( B G D ) )

Syntax hints:   -> wi 4   = wceq 1364  (class class class)co 5925

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rex 2481  df-v 2765  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-iota 5220  df-fv 5267  df-ov 5928

This theorem is referenced by:  csbov123g  5964  prdsplusgfval  12986  prdsmulrfval  12988  issgrp  13105  sgrp1  13113  issgrpd  13114  ismndd  13139  grpsubfvalg  13247  grpsubpropdg  13306  imasgrp  13317  subgsub  13392  releqgg  13426  eqgex  13427  eqgfval  13428  isrng  13566  isrngd  13585  issrg  13597  srgidmlem  13610  isring  13632  ringass  13648  ringidmlem  13654  isringd  13673  ring1  13691  unitlinv  13758  unitrinv  13759  dvrfvald  13765  islmodd  13925  islidlm  14111  rnglidlmsgrp  14129  rnglidlrng  14130  psrval  14296