I knew that $\mathbb{P}(A\cap B|C)=\mathbb{P}(A|C)\cdot \mathbb{P}(B|C)$. However I found (pg. 52, cap II - Dall'Aglio Giorgio, Calcolo delle probabilità) that $\mathbb{P}(A\cap B|C)=\mathbb{P}(A|B)\cdot \mathbb{P}(B|C)$.
What is the difference between these two properties? When should I use the first and when the second?
Thanks in advance for any help!