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This is an except from my textbook (Discrete Mathematics and Its Applications 7th Edition) enter image description here

This was my initial stab at the problem enter image description here (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.

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Yes. Your solution is also acceptable. They are logically equivalent. You have just written your answer in Prenex Normal Form. Furthermore, every sentence in FOL is logically equivalent to a sentence with the quantifiers at the beginning.

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  • $\begingroup$ Yeah that wiki article was pretty overwhelming. Basically mines in prenex normal form because i included all the quantifiers in the beginning? $\endgroup$
    – committedandroider
    Dec 27, 2014 at 22:41
  • $\begingroup$ @committedandroider: Haha, that's correct. Many people (including myself) find this form to be the most natural. $\endgroup$
    – Kyle Gannon
    Dec 27, 2014 at 22:43

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