I am considering first order logic without existential quantification (i.e. with $\forall$ as the only quantifiers). Given an arbitrary formula, would moving all the $\forall$ to the "front" of the formula affect the truth value of this formula? For example, are these two formulas below equivalent (the one below produced by moving all the universal quantifications to the front)? Are all such first order logic formulas (those with $\forall$ as the only quantifiers) equivalent when rewritten this way? How about in higher order logic?
$$ (\forall x,P(x)) \to (\forall y,Q(y)) \\ (\forall x, \forall y, P(x) \to Q(y)) $$