Theorem coeq1d 4883

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 4879	. 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 1395   o. ccom 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-in 3203  df-ss 3210  df-br 4084  df-opab 4146  df-co 4728

This theorem is referenced by:  coeq12d  4886  fcof1o  5913  mapen  7007  hashfacen  11058  znval  14600  znle2  14616