Theorem coeq2d 4840

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 4836	. 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 1373   o. ccom 4679

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-in 3172  df-ss 3179  df-br 4045  df-opab 4106  df-co 4684

This theorem is referenced by:  coeq12d  4842  relcoi1  5214  f1ococnv1  5551  funcoeqres  5553  fcof1o  5858  foeqcnvco  5859  mapen  6943  hashfacen  10981  prdsex  13101  prdsval  13105