Theorem coeq2d 4917

Description: Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Hypothesis

Ref	Expression
coeq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
coeq2d	⊢ (𝜑 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))

Proof of Theorem coeq2d

Step	Hyp	Ref	Expression
1		coeq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		coeq2 4913	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
3	1, 2	syl 14	1 ⊢ (𝜑 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))

Syntax hints:   → wi 4   = wceq 1398   ∘ ccom 4753

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3217  df-ss 3224  df-br 4110  df-opab 4172  df-co 4758

This theorem is referenced by:  coeq12d  4919  relcoi1  5294  f1ococnv1  5643  funcoeqres  5645  fcof1o  5962  foeqcnvco  5963  mapen  7099  hashfacen  11208  prdsex  13482  prdsval  13486  gfsumval  16862  gfsump1  16868