Suppose we have the following inequalities
$$P(A\mid C) > P(B\mid C) \text{ and } P(A\mid \bar{C})>P(B\mid \bar{C})$$
Can we make a conclusion about the following inequality
$$ P(A)>P(B)$$
I have been wondering this question for a while and I tried to apply the Total Average Probability law to it but it didn't get me anywhere.