Theorem coeq1d 3291

Description: Equality deduction for composition of two classes.

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 3287	. 2 |- (A = B -> (A o. C) = (B o. C))
3	1, 2	syl 10	1 |- (ph -> (A o. C) = (B o. C))

Syntax hints:   -> wi 3   = wceq 958   o. ccom 3180

This theorem is referenced by:  mapenlem1 4495  mapenlem2 4496  seq1val 6313  imsval 8312  hocsubdirt 9706  kbass2t 10045  kbass5t 10048

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-br 2625  df-opab 2672  df-co 3193

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