Is "distributing" a quantifier logically equivalent?

Question

Suppose I have this statement, $$∀a:∀b:P(a)→Q(b)$$ and then I have this statement $$(∀a:P(a))→(∀b:Q(b))$$ Are these two statements the same? While I'm not sure about a proof, nor am I sure about my explanation, but I feel like intuitively you can "distribute" the quantifiers over the predicates because those aren't free variables, you are just declaring the quantifier later. Am I right?