Theorem coeq2d 4750

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 4746	. 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 1335   o. ccom 4592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115  df-br 3968  df-opab 4028  df-co 4597

This theorem is referenced by:  coeq12d  4752  relcoi1  5119  f1ococnv1  5445  funcoeqres  5447  fcof1o  5741  foeqcnvco  5742  mapen  6793  hashfacen  10718