I have a random variable $x \in [a, b]$ with PDF $f(x)$ and an event $E$ which satisfies the following property for any $x'<b$.

$$\Pr[E\mid x > x'] \geq \Pr[E]$$

My question is whether or not the following inequality holds.

$$\int_a^b uf(u)\Pr[E\mid x=u] \, du \geq \Pr[E]\int_a^b uf(u) \, du$$