Theorem inin

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

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

Proof

Step	Hyp	Ref	Expression
1		in13	⊢ ( 𝐴 ∩ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐵 ∩ ( 𝐴 ∩ 𝐴 ) )
2		inidm	⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴
3	2	ineq2i	⊢ ( 𝐵 ∩ ( 𝐴 ∩ 𝐴 ) ) = ( 𝐵 ∩ 𝐴 )
4		incom	⊢ ( 𝐵 ∩ 𝐴 ) = ( 𝐴 ∩ 𝐵 )
5	1 3 4	3eqtri	⊢ ( 𝐴 ∩ ( 𝐴 ∩ 𝐵 ) ) = ( 𝐴 ∩ 𝐵 )