Let $A,B,C$ be random variables and suppose $A \perp\!\!\!\perp B$ and $A \perp\!\!\!\perp C$, but that $B$ and $C$ are not independent. Is it true that $A \perp\!\!\!\perp B | C$? I suspect that the answer is no, but I'm struggling to find an example that verifies this.
Any help/hints will is appreciated.