Theorem coeq1d 4827

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 4823	. 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 1364   o. ccom 4667

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-br 4034  df-opab 4095  df-co 4672

This theorem is referenced by:  coeq12d  4830  fcof1o  5836  mapen  6907  hashfacen  10928  znval  14192  znle2  14208