Prove or disprove: $P(A\mid B,C)=P(A\mid C)\rightarrow P(A\mid B)=P(A)$

Question

In my lecture we were given the easy task to prove or disprove

$P(A\mid B,C)=P(A\mid C)\rightarrow P(A\mid B)=P(A)$

and similarly

$P(A\mid B,C)=P(A\mid C)\leftarrow P(A\mid B)=P(A)$

which should be fairly easy, but I cant show it mathematically. I'd very glad for a hint (not a full solution). I'm sure there is one little but critical transformation, which I miss.

Answer 1

Both are false. For the first question use the hint by Matthew Daly. For the second, take $A$ and $B$ independent and $C=(A\cap B)^{c}$. What properties of $P(A)$ and $P(B)$ do you need to get a contradiction?