I need help proving that, if $L$ is a lattice with the property that for every nonempty proper filter $F$ and every ideal $I$ such that $F \bigcap I = \emptyset$ then there is a prime filter $G$ such that $F \subseteq G$ and $I \bigcap G =\emptyset$, then $L$ is distributive.
So, I am a bit confused as to where to even start. What I need to do is prove that for all the elements in $L$, if $a\le b \Rightarrow b' \le a'$, where $a', b'$ are the complements. I don't really know where to start, I have a vague idea of using contraposition, and assuming that L is not distributive, so I could make something like $a \le b$ and $a' \le b'$, so if $a \in F$ then $b \in F$, and so are the complements, but then I $I \bigcap G =\emptyset$ doesn't hold... but I don't know how to justify that. I would really appreciate any input on this, and suggestions on the proper way to write it. Thank you.