Conjunction law for filter quantifiers (${\forall }_{\mathcal{F}}(p\wedge q)⟺({\forall }_{\mathcal{F}}p)\wedge ({\forall }_{\mathcal{F}}q)$)

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.