Theorem coeq1i 5798

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

Hypothesis

Ref	Expression
coeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
coeq1i	⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

Proof of Theorem coeq1i

Step	Hyp	Ref	Expression
1		coeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		coeq1 5796	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

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  cocnvcnv1  6205  ttrclco  9608  hashgval  14240  imasdsval2  17420  prds1  20241  pf1mpf  22267  upxp  23538  uptx  23540  hoico2  31737  hoid1ri  31770  nmopcoadj2i  32082  pjclem3  32177  cycpmconjslem1  33123  cycpmconjs  33125  cyc3conja  33126  1arithidomlem2  33501  erdsze2lem2  35248  pprodcnveq  35925  diblss  41217  cononrel2  43636  trclubgNEW  43659  cortrcltrcl  43781  corclrtrcl  43782  cortrclrcl  43784  cotrclrtrcl  43785  cortrclrtrcl  43786  neicvgbex  44153  neicvgnvo  44156  dvsinax  45959