Logical implications between $\forall x (P(x) \to Q(x))$ and $\forall x P(x) \to \forall x Q(x)$

Question

I am given the following two statements:

$$ 1. ~~(\forall x (P(x) \to Q(x))) \to (\forall x P(x) \to \forall x Q(x)). $$ $$ 2.~~(\forall x P(x) \to \forall x Q(x)) \to (\forall x (P(x) \to Q(x))). $$

Which of these statements is true, and which is false, if any? If the statement is false, could a simple counterexample be provided?

I was guessing that the first statement was false and the second true, reason being that the $\forall$ operator does not distribute over $\vee$ (rather, distributes over $\wedge$), and $P\to Q$ is equivalent to $\neg P \vee Q$. However, I failed to come up with any concrete counterexamples, nor could I provide reasoning as to why the second statement is true, if it even is.

Answer 1

Writing the expressions in plain english makes things clear:

1. If $P(x)$ holds, then $Q(x)$ holds. (This for all $x$)

2. If the statement $P$ is always true, then $Q$ is also always true.

Clearly $1$ implies $2$ but not conversely.