Theorem coeq1d 4695

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
coeq1d	|- ( ph -> ( A o. C ) = ( B o. C ) )

Proof of Theorem coeq1d

Step	Hyp	Ref	Expression
1		coeq1d.1	. 2 |- ( ph -> A = B )
2		coeq1 4691	. 2 |- ( A = B -> ( A o. C ) = ( B o. C ) )
3	1, 2	syl 14	1 |- ( ph -> ( A o. C ) = ( B o. C ) )

Syntax hints:   -> wi 4   = wceq 1331   o. ccom 4538

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-in 3072  df-ss 3079  df-br 3925  df-opab 3985  df-co 4543

This theorem is referenced by:  coeq12d  4698  fcof1o  5683  mapen  6733  hashfacen  10572