Theorem coeq12d 4500

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 4497	. 2 |- ( ph -> ( A o. C ) = ( B o. C ) )
3		coeq12d.2	. . 3 |- ( ph -> C = D )
4	3	coeq2d 4498	. 2 |- ( ph -> ( B o. C ) = ( B o. D ) )
5	2, 4	eqtrd 2072	1 |- ( ph -> ( A o. C ) = ( B o. D ) )

Syntax hints:   -> wi 4   = wceq 1243   o. ccom 4349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-in 2924  df-ss 2931  df-br 3765  df-opab 3819  df-co 4354

This theorem is referenced by: (None)