Theorem oveq123d 6049

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 6045	. 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 6046	. 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 1398  (class class class)co 6028

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-fv 5341  df-ov 6031

This theorem is referenced by:  csbov123g  6067  prdsplusgfval  13447  prdsmulrfval  13449  issgrp  13566  sgrp1  13574  issgrpd  13575  ismndd  13600  grpsubfvalg  13708  grpsubpropdg  13767  imasgrp  13778  subgsub  13853  releqgg  13887  eqgex  13888  eqgfval  13889  isrng  14028  isrngd  14047  issrg  14059  srgidmlem  14072  isring  14094  ringass  14110  ringidmlem  14116  isringd  14135  ring1  14153  unitlinv  14221  unitrinv  14222  dvrfvald  14228  islmodd  14389  islidlm  14575  rnglidlmsgrp  14593  rnglidlrng  14594  psrval  14762