Theorem coeq1i 5314

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 5312	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

Syntax hints:   = wceq 1523   ∘ ccom 5147

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-in 3614  df-ss 3621  df-br 4686  df-opab 4746  df-co 5152

This theorem is referenced by:  coeq12i  5318  cocnvcnv1  5684  hashgval  13160  imasdsval2  16223  prds1  18660  pf1mpf  19764  upxp  21474  uptx  21476  hoico2  28744  hoid1ri  28777  nmopcoadj2i  29089  pjclem3  29184  erdsze2lem2  31312  pprodcnveq  32115  diblss  36776  cononrel2  38218  trclubgNEW  38242  cortrcltrcl  38349  corclrtrcl  38350  cortrclrcl  38352  cotrclrtrcl  38353  cortrclrtrcl  38354  neicvgbex  38727  neicvgnvo  38730  dvsinax  40445