$P\rightarrow (Q\rightarrow R)$ in first-order logic

Question

For statements $P$, $Q$, and $R$, by propositional logic, we have $P\rightarrow (Q\rightarrow R)$ propositionally equivalent to $Q\rightarrow (P\rightarrow R)$.

Will it also be true (via logical equivalence) in first-order logic as long as we take care of the scopes of the bound variables, if any, involved in $P$, $Q$, and/or $R$?

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Edit:
In particular are the following two logically equivalent? \begin{align} &\forall y(P(y)\rightarrow \forall x(Q(x)\rightarrow R(x,y)))\text{; and,}\\ &\forall x(Q(x)\rightarrow \forall y(P(y)\rightarrow R(x,y))).\\ \end{align}

Answer 1

If you really have $P\to (Q\to R)$ etc, then the scopes of the bound variables take care of themselves -- all quantifiers will be either outside the entire formula, or internal to one of $P, Q, R$.

However, if you have something like $P\to \forall x.(Q\to R)$ then you cannot necessarily use the rewriting -- because that does not have the form $P\to(Q\to R)$ in the first place. When you replace propositional variables with entire first-order formulas, that doesn't give you any license to jam quantifiers in between the connectives at other places in the parse tree than the leaves.