 Conjunction law for filter quantifiers

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.