Theorem coeq2d 4818

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 4814	. 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 1364   o. ccom 4659

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-in 3159  df-ss 3166  df-br 4030  df-opab 4091  df-co 4664

This theorem is referenced by:  coeq12d  4820  relcoi1  5189  f1ococnv1  5521  funcoeqres  5523  fcof1o  5824  foeqcnvco  5825  mapen  6893  hashfacen  10897  prdsex  12870