Theorem inin 32020

Description: Intersection with an intersection. (Contributed by Thierry Arnoux, 27-Dec-2016.)

Assertion

Ref	Expression
inin	⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵)

Proof of Theorem inin

Step	Hyp	Ref	Expression
1		in13 4221	. 2 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐵 ∩ (𝐴 ∩ 𝐴))
2		inidm 4217	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	ineq2i 4208	. 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐴)) = (𝐵 ∩ 𝐴)
4		incom 4200	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
5	1, 3, 4	3eqtri 2762	1 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1539   ∩ cin 3946

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-rab 3431  df-v 3474  df-in 3954

This theorem is referenced by:  measinb2  33519