Theorem coeq12d 4855

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 4852	. 2 |- ( ph -> ( A o. C ) = ( B o. C ) )
3		coeq12d.2	. . 3 |- ( ph -> C = D )
4	3	coeq2d 4853	. 2 |- ( ph -> ( B o. C ) = ( B o. D ) )
5	2, 4	eqtrd 2239	1 |- ( ph -> ( A o. C ) = ( B o. D ) )

Syntax hints:   -> wi 4   = wceq 1373   o. ccom 4692

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-in 3176  df-ss 3183  df-br 4055  df-opab 4117  df-co 4697

This theorem is referenced by:  znval  14483  znle2  14499