$P(A \mid B) \leq P(A)$ if and only if $P(A \mid \overline{B}) \geq P(A)$

Question

I am reading a paper that claims (without proof)

$$P(A \mid B) \leq P(A)$$ if and only if $$P(A \mid \overline{B}) \geq P(A)$$

for any two events $A$ and $B$.

This seems reasonable, but I can't seem to prove it directly from the definition of conditional probability. Perhaps there is some identity involving these terms that I'm forgetting?

Answer 1

Hint: $P(A)=P(B)P(A|B) + P(\bar B)P(A|\bar B)$.