Intermediate subgroup condition equals upper intersection-closed

Statement
The following are equivalent for a subgroup property $$p$$:


 * $$p$$ satisfies the intermediate subgroup condition: whenever a subgroup $$H$$ of a group $$G$$ satisfies property $$p$$ in $$G$$, $$H$$ also satisfies property $$p$$ in any intermediate subgroup $$K$$.
 * $$p$$ is closed under finite upper intersections: If $$H \le G$$ and $$K_1, K_2$$ are intermediate subgroups of $$G$$ containing $$H$$, such that $$H$$ satisfies property $$p$$ in both $$K_1$$ and $$K_2$$, then $$H$$ satisfies property $$p$$ in $$K_1 \cap K_2$$.
 * $$p$$ is closed under arbitrary upper intersections.