Theorem coeq12i 5817

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 5813	. 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)
3		coeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	coeq2i 5814	. 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷)
5	2, 4	eqtri 2752	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)

Syntax hints:   = wceq 1540   ∘ ccom 5635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ss 3928  df-br 5103  df-opab 5165  df-co 5640

This theorem is referenced by:  madetsumid  22324  mdetleib2  22451  imsval  30587  pjcmul1i  32103  coprprop  32595  cotrcltrcl  43687  brtrclfv2  43689  clsneif1o  44066  cofuoppf  49112