Using the definition of an
D5811: Event-conditional probability, we have
\begin{equation}
\mathbb{P}(E \mid F)
= \frac{\mathbb{P}(E \cap F)}{\mathbb{P}(F)}
\end{equation}
and thus $\mathbb{P}(E \cap F) = \mathbb{P}(E \mid F) \mathbb{P}(F)$. But now we can switch the places of $E$ and $F$ to obtain
\begin{equation}
\mathbb{P}(E \mid F) \mathbb{P}(F)
= \mathbb{P}(E \cap F)
= \mathbb{P}(F \mid E) \mathbb{P}(E)
\end{equation}
This is precisely what was required to be shown. $\square$
