Complete divisibility-closedness is strongly intersection-closed

Statement
Suppose $$G$$ is a group and $$H_i, i \in I$$ are all completely divisibility-closed subgroups of $$G$$. Then, the intersection of subgroups $$H = \bigcap_{i \in I} H_i$$ is also completely divisibility-closed.

Here, a subgroup is completely divisibility-closed if for any prime number $$p$$ such that every element of the group has a $$p^{th}$$ root in the group, all $$p^{th}$$ roots of any element in the subgroup are in the subgroup.

Related facts

 * Divisibility-closedness is not finite-intersection-closed
 * Powering-invariance is strongly intersection-closed

Proof
Given: A group $$G$$, completely divisibility-closed subgroups $$H_i, i \in I$$ of $$G$$. A prime number $$p$$ such that $$G$$ is $$p$$-divisible. An element $$g \in H = \bigcap_{i \in I} H_i$$. An element $$x \in G$$ such that $$x^p = g$$.

To prove: $$x \in H$$

Proof: It suffices to demonstrate the last sentence, because the existence of $$p^{th}$$ roots in $$G$$ is guaranteed by $$G$$ being $$p$$-divisible.