Theorem coeq2i 4799

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 4797	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)

Syntax hints:   = wceq 1363   ∘ ccom 4642

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-in 3147  df-ss 3154  df-br 4016  df-opab 4077  df-co 4647

This theorem is referenced by:  coeq12i  4802  cocnvcnv2  5152  co01  5155  cocnvres  5165  fcoi1  5408  dftpos2  6275  tposco  6289