# determining decidability with intersection of context free languages

I am trying to solve this problem: Given context-free languages A, B, and C find if the language $$(A\cap B)\cup (B\cap C)$$ is empty. Is this problem decidable.

I know the CFG is not closed under intersection so we do not know that $$A\cap B$$ is also CFG. If it was CFG, I know how to prove decidability. Since we can’t determine about $$A\cap B$$, is there a way to prove whether or not this is decidable?