Theorem coeq12d 4518

Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Hypotheses

Ref	Expression
coeq12d.1	|- ( ph -> A = B )
coeq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
coeq12d	|- ( ph -> ( A o. C ) = ( B o. D ) )

Proof of Theorem coeq12d

Step	Hyp	Ref	Expression
1		coeq12d.1	. . 3 |- ( ph -> A = B )
2	1	coeq1d 4515	. 2 |- ( ph -> ( A o. C ) = ( B o. C ) )
3		coeq12d.2	. . 3 |- ( ph -> C = D )
4	3	coeq2d 4516	. 2 |- ( ph -> ( B o. C ) = ( B o. D ) )
5	2, 4	eqtrd 2113	1 |- ( ph -> ( A o. C ) = ( B o. D ) )

Syntax hints:   -> wi 4   = wceq 1284   o. ccom 4367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-in 2979  df-ss 2986  df-br 3786  df-opab 3840  df-co 4372

This theorem is referenced by: (None)