Conjunction law for filter quantifiers
= Conjunction law for filter quantifiers
{title2=$\forall_{\mathcal F}(p\wedge q)\Longleftrightarrow(\forall_{\mathcal F}p)\wedge(\forall_{\mathcal F}q)$}
The <filter quantifier> satisfies $\forall_{\mathcal F}(p\wedge q)$ if and only if both $\forall_{\mathcal F}p$ and $\forall_{\mathcal F}q$. Finite-intersection closure gives one direction and upward closure gives the other. Properness ensures contradictory truth sets cannot both be filter members.