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

Of course you can prove them, starting from the definition of intersection of sets :

$x \in A \cap B$ iff $x \in A$ and $x \in B$

and the "basic fact" that equality between sets amounts to mutual inclusion : $A = B$ iff $A \subseteq B$ and $B \subseteq A$.
In turn, to prove inclusion you have to use its definition :

$A \subseteq B$ iff : if $x \in A$, then $x \in B$.

You can try yourself the above "machinery" proving that :


$(A \cap B) \cap (B \cap C) = A \cap B \cap C$.


