Theorem coeq1i 5729

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 5727	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)

Syntax hints:   = wceq 1533   ∘ ccom 5558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-in 3942  df-ss 3951  df-br 5066  df-opab 5128  df-co 5563

This theorem is referenced by:  coeq12i  5733  cocnvcnv1  6109  hashgval  13692  imasdsval2  16788  prds1  19363  pf1mpf  20514  upxp  22230  uptx  22232  hoico2  29533  hoid1ri  29566  nmopcoadj2i  29878  pjclem3  29973  cycpmconjslem1  30796  cycpmconjs  30798  cyc3conja  30799  erdsze2lem2  32451  pprodcnveq  33344  diblss  38305  cononrel2  39953  trclubgNEW  39976  cortrcltrcl  40083  corclrtrcl  40084  cortrclrcl  40086  cotrclrtrcl  40087  cortrclrtrcl  40088  neicvgbex  40460  neicvgnvo  40463  dvsinax  42195