Which probability law? It may be a basic probability law in another form, but I cannot figure it out. Why can we say the following:
$P(A∩B|C) = $$P(A|B∩C)P(B|C)$
Thank you.

user76553, in your most recent comment you asked "If I did not know the rhs of the equation, how could I derive it from the lhs? Specifically, what probability equations would I use?" 
I'm assuming you mean if you just had $P(A\cap B \mid C)$ sitting in front of you by itself without knowing that it was indeed equivalent to $P(A\mid B \cap C)P(B \mid C)$, and you wanted to see if you could find equivalent expressions. 
In that case, you still make use of the various equalities given by marchimedes in his/her answer above, but you could proceed like this: 
Step 1: use definition of conditional probability to write
$$P(A \cap B \mid C) = \frac{P(A \cap B \cap C)}{P(C)} $$
Step 2: use the definition of conditional probability again on the numerator to express $P(A \cap B \cap C)$ as $P(A \mid B \cap C)P(B \cap C)$. Then the entire expression is now
$$\frac{P(A \mid B \cap C)P(B \cap C)}{P(C)} $$
Step 3: similarly we then express $P(B \cap C)$ as $P(B \mid C)P(C)$ and the entire expression is now
$$\frac{P(A \mid B \cap C)P(B \mid C)P(C)}{P(C)} $$
Step 4: cancel the $P(C)$ terms in the numerator and denominator and then you have
$$P(A \mid B \cap C)P(B \mid C).$$
So really you are just using the definition of conditional probability several times and then one cancelation at the end. 
Hope that helps!
