Theorem coeq12d 4803

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 4800	. 2 |- ( ph -> ( A o. C ) = ( B o. C ) )
3		coeq12d.2	. . 3 |- ( ph -> C = D )
4	3	coeq2d 4801	. 2 |- ( ph -> ( B o. C ) = ( B o. D ) )
5	2, 4	eqtrd 2220	1 |- ( ph -> ( A o. C ) = ( B o. D ) )

Syntax hints:   -> wi 4   = wceq 1363   o. ccom 4642

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-in 3147  df-ss 3154  df-br 4016  df-opab 4077  df-co 4647

This theorem is referenced by: (None)