Distributing Set Intersections Over an Intersection I was working through some examples, and found this to be true: 
$(A \cap B) \cap (B \cap C) = A \cap B \cap C $
$(A \cap B) \cap(A \cap C) = A \cap B \cap C$
$(A \cap B) \cap(A \cap C) \cap (B \cap C) = A \cap B \cap C$
Is there some proof, or rule for these statements to hold true?
Thank you

Note:
$$
A \cap B = B \cap A
$$
so that all three statements are related. Now we can prove the first one (and thus all three) through double inclusion i.e. we prove
$$
(A \cap B) \cap (B \cap C) \subset A \cap B \cap C \text{ and } (A \cap B) \cap (B \cap C) \supset A \cap B \cap C
$$
which shows
$$
(A \cap B) \cap (B \cap C) = A \cap B \cap C
$$
Do you know how to show the above?
Spoiler:

 $x \in (A \cap B) \cap (B \cap C) \implies x \in A, B, C \implies x \in A \cap B \cap C$. $x \in A \cap B \cap C \implies x \in A,B , x \in B,C \implies x \in (A \cap B) \cap (B \cap C)$

