I just wanted to ask about this equality

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

Can this easily be proven by using the associative property of sets which states?

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

Or is this not a valid mathematical proof