Probability conditioned on two independent events

Question

Let $B$ and $C$ be independent events. We are interested in $P[A|B \cap C]$. We know what $P[A|B]$ and $P[A|C]$ are. How do we use these two to find the answer? What additional missing pieces, if any, do we need?

I arrived at $$ P[A|B \cap C] = \frac{P[A \cap B \cap C]}{P[B]P[C]}. $$

But I have not been able to make use of the given quantities in some way.

Answer 1

P(A|B,C)=P(A,B|C)/P(B|C). So if A and B are conditionally independent given C, then the desired probability equals the given probability P(A|C). A similar argument holds for if A and C are conditionally independent given B.