Theorem coeq2i 5809

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 5807	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)

Syntax hints:   = wceq 1541   ∘ ccom 5628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ss 3918  df-br 5099  df-opab 5161  df-co 5633

This theorem is referenced by:  coeq12i  5812  cocnvcnv2  6217  co01  6220  dfpo2  6254  fcoi1  6708  f1ofvswap  7252  dftpos2  8185  tposco  8199  cottrcl  9628  canthp1  10565  cats1co  14779  isoval  17689  mvdco  19374  evlsval  22041  evl1fval1lem  22274  evl1var  22280  pf1ind  22299  rhmply1vr1  22331  rhmply1vsca  22332  imasdsf1olem  24317  hoico1  31831  hoid1i  31864  pjclem1  32270  pjclem3  32272  pjci  32275  cycpmconjv  33224  cycpmconjs  33238  poimirlem9  37830  cdlemk45  41217  cononrel1  43845  trclubgNEW  43869  trclrelexplem  43962  relexpaddss  43969  cotrcltrcl  43976  cortrcltrcl  43991  corclrtrcl  43992  cotrclrcl  43993  cortrclrcl  43994  cotrclrtrcl  43995  cortrclrtrcl  43996  brco3f1o  44284  clsneibex  44353  neicvgbex  44363  subsaliuncl  46612  meadjiun  46720  fundcmpsurinjimaid  47667  dftpos5  49129  tposrescnv  49134