Suppose that the following identity holds. $$P(A|B)=P(A|B \cap C)P(C)+P(A|B \cap C^C)P(C^C)$$ Also assume that, $P(A|B \cap C) \neq P(A|B)$ and $P(A)>0$. Then we have to show that $B$ and $C$ are independent events, i.e. $P(B \cap C)=P(B)P(C)$.
Note : The original problem was deriving the identity, assuming independence of $B$ and $C$, which can be shown easily. I'm stuck at this converse version.
Source : Rohatgi, Saleh - Problem $9$, page $36$.
Any help would be much appreciated. Thank you.