Theorem inin 32586

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 4171	. 2 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐵 ∩ (𝐴 ∩ 𝐴))
2		inidm 4167	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	ineq2i 4157	. 2 ⊢ (𝐵 ∩ (𝐴 ∩ 𝐴)) = (𝐵 ∩ 𝐴)
4		incom 4149	. 2 ⊢ (𝐵 ∩ 𝐴) = (𝐴 ∩ 𝐵)
5	1, 3, 4	3eqtri 2763	1 ⊢ (𝐴 ∩ (𝐴 ∩ 𝐵)) = (𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1542   ∩ cin 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-in 3896

This theorem is referenced by:  measinb2  34367