Theorem coeq1d 4878

Description: Equality deduction for composition of two classes. (Contributed by set.mm contributors, 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 4874	. 2 |- (A = B -> (A o. C) = (B o. C))
3	1, 2	syl 15	1 |- (ph -> (A o. C) = (B o. C))

Syntax hints:   -> wi 4   = wceq 1642   o. ccom 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-opab 4623  df-br 4640  df-co 4726

This theorem is referenced by:  coeq12d  4881  enmap1lem3  6071  enmap1lem5  6073