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?
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Referring to the sentences marked (#) in my recent answer here, notice that distributing the quantifiers around the conditional $(→)$ doesn't quite work as you think. Click on the blue link "Prenex form" for wikipedia's explanation. Re: your current question: the two sentences aren't equivalent to each other, unfortunately: if every $a$ satisfies the rest of your first sentence, then in fact, some $a$ satisfying $P$ is sufficient for every $b$ to satisfy $Q.$ – ryang Feb 18 '22 at 05:50