Logical equivalence of quantified statements

Forward. If every $x$ satisfies both $P$ and $Q$, then every $x$ satisfies $P$, and every $x$ satisfies $Q$.

Backward. If every $x$ satisfies $P$, and every $x$ satisfies $Q$, then for any $x$, both $P(x)$ and $Q(x)$ hold.

Conclusion. Yes, the equivalence holds.

Contrast. $\forall x,\ (P(x)\lor Q(x))$ is *not* equivalent to $(\forall x,\ P(x))\lor(\forall x,\ Q(x))$.