Theorem coeq1i 5485

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 5483	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

Syntax hints:   = wceq 1653   ∘ ccom 5316

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-in 3776  df-ss 3783  df-br 4844  df-opab 4906  df-co 5321

This theorem is referenced by:  coeq12i  5489  cocnvcnv1  5865  hashgval  13373  imasdsval2  16491  prds1  18930  pf1mpf  20038  upxp  21755  uptx  21757  hoico2  29141  hoid1ri  29174  nmopcoadj2i  29486  pjclem3  29581  erdsze2lem2  31703  pprodcnveq  32503  diblss  37191  cononrel2  38684  trclubgNEW  38708  cortrcltrcl  38815  corclrtrcl  38816  cortrclrcl  38818  cotrclrtrcl  38819  cortrclrtrcl  38820  neicvgbex  39192  neicvgnvo  39195  dvsinax  40871