Joint probability of three events, when two of them are independent

Question

My main question is: if B and C are independent, then can we conclude the following:
P(A|B,C)=P(A|B)*P(A|C)?
Simply it can be seen that the conclusion is true if and only if:
A and B independent => P(A,B,C)=P(A,B)*P(A,C)
Now I'm wondering whether it's true or not.
Thanks.

Answer 1

Hint: Here is a counterexample. Toss a fair coin twice. Let $B$ be the event the first toss gives head, let $C$ be the event the second toss gives head. Let $A$ be the event both give head.

Then $\Pr(A|B,C)=1$ while $\Pr(A|B)\Pr(A|C)=\frac{1}{4}$.