SOLVED If A and B are independent given C, then P(A|C) x P(B|C) = P(A and B | C)?

I think my professor used this property to solve some excersises in my University, but I can't find it in my notes nor recommended textbook.

Is this true or false:

If $A$ and $B$ are independent under C (ie, if $C$ occurs then $A$ and $B$ are independent; no matter if they were or not independent before $C$), then $$P((A \cap B) | C) = P(A|C) \times P(B|C).$$

As others have already mentioned, if $A$ and $B$ are conditionally independent given an event $C$ and $P(C)\not =0$, then we have $$P(A \cap B \mid C)=P(A\mid C)\cdot P(B\mid C)$$ $$=\frac{P(A\cap C)\cdot P(B\cap C)}{P(C)\cdot P(C)}$$ However in your post, it isn't exactly clear if $P(C)\not =0$, so we can't say for sure.