Theorem coeq12i 5850

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 5846	. 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)
3		coeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	coeq2i 5847	. 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷)
5	2, 4	eqtri 2792	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)

Syntax hints:   = wceq 1567   ∘ ccom 5666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ss 3930  df-br 5114  df-opab 5178  df-co 5671

This theorem is referenced by:  madetsumid  22586  mdetleib2  22713  imsval  30977  pjcmul1i  32493  coprprop  32984  cotrcltrcl  44342  brtrclfv2  44344  clsneif1o  44721  cofuoppf  49812