Theorem coeq1i 5839

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 5837	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

Syntax hints:   = wceq 1540   ∘ ccom 5658

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ss 3943  df-br 5120  df-opab 5182  df-co 5663

This theorem is referenced by:  coeq12i  5843  cocnvcnv1  6246  ttrclco  9730  hashgval  14349  imasdsval2  17528  prds1  20281  pf1mpf  22288  upxp  23559  uptx  23561  hoico2  31684  hoid1ri  31717  nmopcoadj2i  32029  pjclem3  32124  cycpmconjslem1  33111  cycpmconjs  33113  cyc3conja  33114  1arithidomlem2  33497  erdsze2lem2  35172  pprodcnveq  35847  diblss  41135  cononrel2  43566  trclubgNEW  43589  cortrcltrcl  43711  corclrtrcl  43712  cortrclrcl  43714  cotrclrtrcl  43715  cortrclrtrcl  43716  neicvgbex  44083  neicvgnvo  44086  dvsinax  45890