This is an except from my textbook (Discrete Mathematics and Its Applications 7th Edition)
This was my initial stab at the problem (with domain of both variables being all real numbers)
Would it also be acceptable in my solution to introduce the y variable a bit earlier than the author's solution. What I was thinking was for every real number $x$, that exists a real number $y$ for which the implication will eventually be evaluated for. I understand what the author was getting at though, you don't really care about the $y$ unless you meet the first condition, then you can iterate the $y$s. Would my solution be acceptable as well? I believe they are logically equivalent.