If $P(A\cap B\cap C)=P(A\cap B)P(C)$, does it follow that $P(A\cap C)=P(A)P(C)$ and $P(B\cap C)=P(B)P(C)$? 
If $P(A\cap B\cap C)=P(A\cap B)P(C)$, does it follow that $P(A\cap C)=P(A)P(C)$ and $P(B\cap C)=P(B)P(C)$?

In other words, if the conjunction of two events $A$ and $B$ is independent of a third event $C$, is it always true that event $C$ is independent from events $A$ and $B$ separately?
I've gone back and forth between believing that it is likely true and not true. I can't see how to prove it from the basic axioms of probability, but it also seems challenging to think of a counterexample where the conjunction of $A$ and $B$ is independent of $C$, but either $A$ or $B$ are dependent on $C$ (or vice versa).

Take any dependent events $A$ and $C$ and put $B=A^c$ (complement of $A$) to get a counter example.
