Theorem coeq12i 5810

Description: Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.)

Hypotheses

Ref	Expression
coeq12i.1	⊢ 𝐴 = 𝐵
coeq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
coeq12i	⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)

Proof of Theorem coeq12i

Step	Hyp	Ref	Expression
1		coeq12i.1	. . 3 ⊢ 𝐴 = 𝐵
2	1	coeq1i 5806	. 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)
3		coeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	coeq2i 5807	. 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷)
5	2, 4	eqtri 2757	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)

Syntax hints:   = wceq 1541   ∘ ccom 5626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ss 3916  df-br 5097  df-opab 5159  df-co 5631

This theorem is referenced by:  madetsumid  22403  mdetleib2  22530  imsval  30709  pjcmul1i  32225  coprprop  32727  cotrcltrcl  43908  brtrclfv2  43910  clsneif1o  44287  cofuoppf  49337