Seems to be you want to check whether .

Well, let be the universe from where we get our objects ,and let P(x) = the absolute value of x is x itself (i.e., P(x) iff ,and let Q(x)= x is a non-negative integer (i.e., Q(x) iff ) , then:

means: for any integer x, , which is true, whereas

means: for any integer x we have for any integer x, , which is false.

If I didn't commit some logical mistake above () then the answer is: no, they aren't logical equivalent.

Tonio