Theorem coeq2i 5799

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 5797	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)

Syntax hints:   = wceq 1541   ∘ ccom 5618

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 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ss 3914  df-br 5090  df-opab 5152  df-co 5623

This theorem is referenced by:  coeq12i  5802  cocnvcnv2  6206  co01  6209  dfpo2  6243  fcoi1  6697  f1ofvswap  7240  dftpos2  8173  tposco  8187  cottrcl  9609  canthp1  10545  cats1co  14763  isoval  17672  mvdco  19357  evlsval  22021  evl1fval1lem  22245  evl1var  22251  pf1ind  22270  rhmply1vr1  22302  rhmply1vsca  22303  imasdsf1olem  24288  hoico1  31736  hoid1i  31769  pjclem1  32175  pjclem3  32177  pjci  32180  cycpmconjv  33111  cycpmconjs  33125  poimirlem9  37679  cdlemk45  41056  cononrel1  43697  trclubgNEW  43721  trclrelexplem  43814  relexpaddss  43821  cotrcltrcl  43828  cortrcltrcl  43843  corclrtrcl  43844  cotrclrcl  43845  cortrclrcl  43846  cotrclrtrcl  43847  cortrclrtrcl  43848  brco3f1o  44136  clsneibex  44205  neicvgbex  44215  subsaliuncl  46466  meadjiun  46574  fundcmpsurinjimaid  47521  dftpos5  48984  tposrescnv  48989