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

$(A \cap B) \cap (B \cap C) = A \cap (B \cap B) \cap C  = A \cap B \cap C$, using associativity of $\cap$ (i.e. it doesn't matter where we put brackets), and $B \cap B = B$.
The second on wis similar, but we also use commutativity (the order in $\cap$ does not matter): $(A \cap B) \cap (A \cap C) = A \cap B \cap A \cap C = (A \cap A) \cap B \cap C = A \cap B \cap C$. 
You probably yes how to do the third one this way as well.
