Theorem coeq1i 5808

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 5806	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

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  cocnvcnv1  6216  ttrclco  9627  hashgval  14256  imasdsval2  17437  prds1  20258  pf1mpf  22296  upxp  23567  uptx  23569  hoico2  31832  hoid1ri  31865  nmopcoadj2i  32177  pjclem3  32272  cycpmconjslem1  33236  cycpmconjs  33238  cyc3conja  33239  1arithidomlem2  33617  erdsze2lem2  35398  pprodcnveq  36075  diblss  41430  cononrel2  43836  trclubgNEW  43859  cortrcltrcl  43981  corclrtrcl  43982  cortrclrcl  43984  cotrclrtrcl  43985  cortrclrtrcl  43986  neicvgbex  44353  neicvgnvo  44356  dvsinax  46157