Theorem coeq12i 5843

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 5839	. 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)
3		coeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	coeq2i 5840	. 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷)
5	2, 4	eqtri 2783	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)

This proof depends on syntax axioms:   = wceq 1570   ∘ ccom 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ss 3916  df-br 5104  df-opab 5168  df-co 5664

This theorem is used by:  madetsumid  22684  mdetleib2  22811  imsval  31167  pjcmul1i  32683  coprprop  33172  cotrcltrcl  44566  brtrclfv2  44568  clsneif1o  44945  cofuoppf  50077