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.

You should simply expand upon the definition:
$P(A \wedge B | C) = \frac{P(A \wedge B \wedge C)}{P(C)}$. 
$P(A | B \wedge C) = \frac {P( A \wedge B \wedge C)}{P( B \wedge C )}$.
$P(B|C)=\frac{P( B \wedge C)}{P(C)}$.
Now, we solve for $P(A \wedge B \wedge C)$ in the first two above and equate them:
$P(A \wedge B | C) \cdot P(C)=P(A | B \wedge C) \cdot P( B \wedge C )$.
Solve for $P(B \wedge C)$ in the third line and plug it in to the expression above.
