Theorem coeq2i 4846

Description: Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.)

Hypothesis

Ref	Expression
coeq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
coeq2i	⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)

Proof of Theorem coeq2i

Step	Hyp	Ref	Expression
1		coeq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		coeq2 4844	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)

Syntax hints:   = wceq 1373   ∘ ccom 4687

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-in 3176  df-ss 3183  df-br 4052  df-opab 4114  df-co 4692

This theorem is referenced by:  coeq12i  4849  cocnvcnv2  5203  co01  5206  cocnvres  5216  fcoi1  5468  dftpos2  6360  tposco  6374