Strongly UL-intersection-closed subgroup property

Definition
A subgroup property $$p$$ is termed strongly UL-intersection-closed if for any group $$G$$, any (possibly empty) indexing set $$I$$, and subgroups $$H_i \le K_i \le G, i \in I$$, such that $$H_i$$ satisfies property $$p$$ in $$K_i$$ for each $$i \in I$$, we have:

$$\bigcap_{i \in I} H_i$$ satisfies property $$p$$ in $$\bigcap_{i \in I} K_i$$.

A subgroup property is strongly UL-intersection-closed if and only if it is both UL-intersection-closed and identity-true.

Weaker metaproperties

 * Stronger than::UL-intersection-closed subgroup property
 * Stronger than::Strongly intersection-closed subgroup property
 * Stronger than::Intersection-closed subgroup property
 * Stronger than::Transfer condition
 * Stronger than::Intermediate subgroup condition