Theorem coeq12d 4615

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 4612	. 2 |- ( ph -> ( A o. C ) = ( B o. C ) )
3		coeq12d.2	. . 3 |- ( ph -> C = D )
4	3	coeq2d 4613	. 2 |- ( ph -> ( B o. C ) = ( B o. D ) )
5	2, 4	eqtrd 2121	1 |- ( ph -> ( A o. C ) = ( B o. D ) )

Syntax hints:   -> wi 4   = wceq 1290   o. ccom 4458

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-in 3008  df-ss 3015  df-br 3854  df-opab 3908  df-co 4463

This theorem is referenced by: (None)