Theorem coeq1i 5281

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 5279	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

Syntax hints:   = wceq 1483   ∘ ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-co 5123

This theorem is referenced by:  coeq12i  5285  cocnvcnv1  5646  hashgval  13120  imasdsval2  16176  prds1  18614  pf1mpf  19716  upxp  21426  uptx  21428  hoico2  28616  hoid1ri  28649  nmopcoadj2i  28961  pjclem3  29056  erdsze2lem2  31186  pprodcnveq  31990  diblss  36459  cononrel2  37901  trclubgNEW  37925  cortrcltrcl  38032  corclrtrcl  38033  cortrclrcl  38035  cotrclrtrcl  38036  cortrclrtrcl  38037  neicvgbex  38410  neicvgnvo  38413  dvsinax  40127