Quantifier explination: $\forall x(R\to Q(x))\equiv R \to \forall x (Q(x))$ I don't understand the proof for this problem and couldn't find the answer anywhere.
$$\forall x(R\to Q(x))\equiv R \to \forall x (Q(x))$$

Think of it first from a concrete approach, in order to get a better feel for what you're looking to prove in the general case.
Example: (Domain of x is all people)
$R:$ It is raining. (Note that R is independent of any x)
$Q(x): x$ takes an umbrella or gets wet.
$\forall x(R \rightarrow Q(x))\tag 1$
$R\rightarrow \forall x(Q(x))\tag 2$
$(1)$ "says" Everyone, if it's raining, takes an umbrella or gets wet.
$(2)$ "says" If it is raining, everyone takes an umbrella or gets wet.
The important thing to note is that $R$, by itself, is a statement, and can, but need not be included in the bounds over which x is quantified.
Hopefully this explanation helps you to understand the general case. @Hagen von Eitzen has already given you the technical details of the proof you may be seeking.
In the future, when you encounter a question which overwhelms upon first site, break it down into smaller pieces. If you are being asked to prove anything about $x$, universally (in a given domain)...get a feel for what it looks like concretely,  in a particular interpretation of the domain of $x$ and predicates used, to try to get a sense of what you are working with.
