Maximal (or prime) theories and their sets

Question

Probably it is very simple lemma but I cannot see it. Suppose that we have intuitionistic propositional logic (in fact, it can be classical) and $W$ is the set of all prime (or maximal - in classical setting) theories. Let $|\varphi|$ be the proof set (or the so-called prime set) of formula $\varphi$, defined as such: $|\varphi| = \{z \in W; \varphi \in z\}$. Now the lemma is: if $|\varphi| = |\psi|$ then $\vdash \varphi \leftrightarrow \psi$. I understand that $\varphi$ and $\psi$ are equivalent on the ground of prime theories from $|\varphi|$ and $|\psi|$ but why $\varphi \leftrightarrow \psi$ must be theorem?

Answer 1

Any filter (consistent theory) in distributive lattice is the intersection of the prime filters (prime theories) containing it, by Zorn's lemma. So in your situation, the filter generated by $\phi$ must be the same as the filter generated by $\psi$, because they are intersections of the same thing. (I allow theories generated by statements that are not sentences, but I imagine the argument is very analogous howsoever you define theories.)