Why does $P(A \cap B \mid C) = P(A \mid B\cap C) \cdot P(B \mid C)$? I'm guessing this is some kind of application of the total law of probability and Bayes' theorem, I'm just not exactly sure on the steps of how to show it.
I'm mainly confused by the term $P(A  \mid  B\cap C)$. More specifically, $B\cap C$. Why is it conditioned on $B$ AND $C$?

$$P(A \cap B  \mid  C) = \frac{P(A\cap B \cap C)}{P(C)}
= \frac{P(A\cap B \cap C)}{P(B\cap C)} \frac{P(B \cap C)}{P(C)}
= P(A \mid B\cap C) P(B \mid C)$$
