Theorem coeq1i 5858

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 5856	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

Syntax hints:   = wceq 1539   ∘ ccom 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-br 5148  df-opab 5210  df-co 5684

This theorem is referenced by:  coeq12i  5862  cocnvcnv1  6255  ttrclco  9715  hashgval  14297  imasdsval2  17466  prds1  20211  pf1mpf  22091  upxp  23347  uptx  23349  hoico2  31277  hoid1ri  31310  nmopcoadj2i  31622  pjclem3  31717  cycpmconjslem1  32583  cycpmconjs  32585  cyc3conja  32586  erdsze2lem2  34493  pprodcnveq  35159  diblss  40344  cononrel2  42648  trclubgNEW  42671  cortrcltrcl  42793  corclrtrcl  42794  cortrclrcl  42796  cotrclrtrcl  42797  cortrclrtrcl  42798  neicvgbex  43165  neicvgnvo  43168  dvsinax  44927