Theorem inin

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

	Ref	Expression
Assertion	inin	$$⊢A\cap \left(A\cap B\right)=A\cap B$$

Proof

Step	Hyp	Ref	Expression
1		in13	$$⊢A\cap \left(A\cap B\right)=B\cap \left(A\cap A\right)$$
2		inidm	$$⊢A\cap A=A$$
3	2	ineq2i	$$⊢B\cap \left(A\cap A\right)=B\cap A$$
4		incom	$$⊢B\cap A=A\cap B$$
5	1 3 4	3eqtri	$$⊢A\cap \left(A\cap B\right)=A\cap B$$