Quantifiers and Predicates in Discrete Mathematics I was doing midterm review and I came across these formulas
$$\forall x \big( P(x) \to Q (x))$$
and
$$\forall x P (x) \to \forall x Q (x)$$
I wanted to know what the difference was in terms of $x$ for both of them. In the first one, if I choose $x = 5$, then it's $P(5) \to Q(5)$ but is it the same for the second one? If there are quantifiers on both, does that mean that $x$ can be a different value for both such as $P(5) \to Q(3)$?

The second statement says that if $P(x)$ holds of all $x$, then $Q(x)$ also holds of all $x$.  Unlike the first statement, which would allow us to conclude $Q(5)$ from $P(5)$, the second statement would not allow us to conclude $Q(5)$ from $P(x)$ for any particular $x$.  We would only be able to conclude $Q(5)$ if we knew $P(x)$ held for all $x$.  If $P(x)$ failed to hold for even a single $x$, we would not be able to draw any conclusion about $Q(5)$.  On the other hand, if $P(x)$ held for all $x$, then we could conclude that $Q(x)$ held, not just for $x=5$, but for all $x$.
Note also that the first statement entails the second statement: if $Q(x)$ holds whenever $P(x)$ holds, then if $P(x)$ holds for all $x$, we know that $Q(x)$ holds for all $x$.
