Moving the universal quantifier when some statement is independent of the corresponding variable If $P(x)$ is a predicate about $x$, while $Q$ is a statement, independent of $x$, is the following true?

$(\forall x P(x))\iff Q\equiv \forall x(P(x)\iff Q)$

Please note, I have a very low knowledge about mathematical logic, so I might need some pedagogical explanation. I am just trying to translate some texts into mathematical symbols, in order to prove in different ways. If anyone can confirm this claim without explanation, it's fine with me.
 A: Yes it is equivalent.
In fact this can be seen as a rule of prenex operations: https://en.wikipedia.org/wiki/Prenex_normal_form
The actual rule goes: You can replace $Q x( B )\lor C$ by $Q x(B \lor C)$, and vice-versa, provided that $x$ is not free on $C$ (where $Q$ is any quantifier). You can verify that you can translate the bi-implication in your formula with $\neg, \lor$, then you see that you can put the universal in the left of the parenthesis. Note that your $Q$ (not the quantifier, but the statement you gave) doesn't have any $x$ in it, so $x$ is not free in it (it doesn't have any quantifier bounding it).
Prenex operations are useful to put quantifier in the normal form (to the left of all terms). You can see books on logic for proofs or justification of this rules.
Well, but you can see this actually with another light: if you have a proposition where $x$ don't appear, it means that you can generalize it with a quantifier. Intuitively: "it rained today" is true iff "for all cats, it rained today" is true (not a good example, but it is the one came in my mind hehe). From there you can justify the equivalence.
