Theorem coeq1i 5803

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 5801	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

Syntax hints:   = wceq 1548   ∘ ccom 5624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ss 3901  df-br 5075  df-opab 5137  df-co 5629

This theorem is referenced by:  coeq12i  5807  cocnvcnv1  6212  ttrclco  9634  hashgval  14290  imasdsval2  17475  prds1  20296  pf1mpf  22341  upxp  23609  uptx  23611  hoico2  31848  hoid1ri  31881  nmopcoadj2i  32193  pjclem3  32288  cycpmconjslem1  33237  cycpmconjs  33239  cyc3conja  33240  1arithidomlem2  33629  selvascl  33711  erdsze2lem2  35445  pprodcnveq  36122  diblss  41675  cononrel2  44052  trclubgNEW  44075  cortrcltrcl  44197  corclrtrcl  44198  cortrclrcl  44200  cotrclrtrcl  44201  cortrclrtrcl  44202  neicvgbex  44569  neicvgnvo  44572  dvsinax  46368