Theorem coeq2i 4896

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

Hypothesis

Ref	Expression
coeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
coeq2i	⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)

Proof of Theorem coeq2i

Step	Hyp	Ref	Expression
1		coeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		coeq2 4894	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)

Syntax hints:   = wceq 1398   ∘ ccom 4735

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 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-in 3207  df-ss 3214  df-br 4094  df-opab 4156  df-co 4740

This theorem is referenced by:  coeq12i  4899  cocnvcnv2  5255  co01  5258  cocnvres  5268  fcoi1  5525  dftpos2  6470  tposco  6484