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.

Starting from the right and using just the definition of conditional probability,
$$\begin{align}
P(A\mid B\cap C)P(B\mid C) 
&= \frac{P(A\cap B \cap C)}{P(B\cap C)}\times \frac{P(B\cap C)}{P(C)}
&\scriptstyle{\text{substitute using the definition of conditional probability}}\\
&= \frac{P(A\cap B \cap C)}{P(C)}
&\scriptstyle{\text{cancel}}~P(B\cap C)~\scriptstyle{\text{from numerator and denominator}}\\
&= P(A\cap B \mid C).  &\scriptstyle{\text{identify the ratio as a conditional probability}}
\end{align}$$
Starting from the left, we can just reverse the steps above.
$$\begin{align}
P(A\cap B \mid C) &= \frac{P(A\cap B \cap C)}{P(C)}\\
&= \frac{P(A\cap B \cap C)}{P(B\cap C)}\times \frac{P(B\cap C)}{P(C)}\\
&= P(A\mid B\cap C)P(B\mid C).
\end{align}$$

But you might ask: given the expression $P(A\cap B \mid C)$, how would
I even begin to think of ways to get to $P(A\mid B\cap C)P(B\mid C)$ which expression
I have never seen before and have no reason to suspect might
equal $P(A\cap B \mid C)$?  Well, one reason to suspect that something
like this might be true is to consider that conditional probabilities
conditioned on the occurrence of the same event are a valid probability measure, 
that is,
every theorem of probability, e.g. $$P(A\cup B) = P(A)+P(B)-P(A\cap B),$$
holds if all the probabilities in question are changed
to conditional probabilities
given that event $C$ occurred. Thus,
$$P(A\cup B\mid C) = P(A\mid C)+P(B\mid C)-P(A\cap B\mid C),$$
and similarly,
$$P(A\cap B) = P(A\mid B)P(B) ~\Rightarrow ~
P(A\cap B\mid C) = P(A\mid B\cap C)P(B\mid C).$$
So, once you have the right intuition, you can, if you wish,
formally verify the
correctness of the result using the steps outlined above.
