Theorem coeq1d 4638

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 4634	. 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 1299   o. ccom 4481

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-in 3027  df-ss 3034  df-br 3876  df-opab 3930  df-co 4486

This theorem is referenced by:  coeq12d  4641  fcof1o  5622  mapen  6669  hashfacen  10420