Theorem coeq12i 5424

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 5420	. 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)
3		coeq12i.2	. . 3 ⊢ 𝐶 = 𝐷
4	3	coeq2i 5421	. 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷)
5	2, 4	eqtri 2792	1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)

Syntax hints:   = wceq 1630   ∘ ccom 5253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-in 3728  df-ss 3735  df-br 4785  df-opab 4845  df-co 5258

This theorem is referenced by:  madetsumid  20484  mdetleib2  20611  imsval  27874  pjcmul1i  29394  cotrcltrcl  38536  brtrclfv2  38538  clsneif1o  38921