Theorem coeq2d 4919

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 4915	. 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 1398   o. ccom 4755

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-in 3219  df-ss 3226  df-br 4112  df-opab 4174  df-co 4760

This theorem is referenced by:  coeq12d  4921  relcoi1  5296  f1ococnv1  5645  funcoeqres  5647  fcof1o  5964  foeqcnvco  5965  mapen  7101  hashfacen  11212  prdsex  13499  prdsval  13503  gfsumval  16879  gfsump1  16885