Theorem coeq2d 4516

Description: Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Hypothesis

Ref	Expression
coeq1d.1	|- ( ph -> A = B )

Assertion

Ref	Expression
coeq2d	|- ( ph -> ( C o. A ) = ( C o. B ) )

Proof of Theorem coeq2d

Step	Hyp	Ref	Expression
1		coeq1d.1	. 2 |- ( ph -> A = B )
2		coeq2 4512	. 2 |- ( A = B -> ( C o. A ) = ( C o. B ) )
3	1, 2	syl 14	1 |- ( ph -> ( C o. A ) = ( C o. B ) )

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:  coeq12d  4518  relcoi1  4869  f1ococnv1  5175  funcoeqres  5177  fcof1o  5449  foeqcnvco  5450