Does the position of ∃ matter?

Question

Everyone who makes an A aces some final exam

Is this

∀x,∃y: makes(x, A) => aces(x, y)

the same as

∀x: makes(x, A) => ∃y: aces(x, y)

where x represents everyone and y represents exams.

Does the position of ∃ matter?

Answer 1

No, they are equivalent.

This is part of a theorem called "Prenex normal form".

Answer 2

Let $M=\{x\mid\text{makes}(x,A)\}$ and $E$ the set of exams. Then, both assertions are equivalent to: $$ \forall x\in M,\quad\exists y\in E,\quad\text{aces}(x,y). $$