Logic - Propositions Proof

I'm not sure what your difficulty is. The point is that the statement "for all y, there exist x such that P(x,y) is true" is NOT equivalent to "there exist x such that, for all y, P(x,y) is true". They prove that by giving an example statement, P(x, y)= "x> y".

It is certainly true that "For all y, there exist x such that x> y". For any y, for example, x= y+ 1 is larger than y.

But it is NOT true that "There exist x such that, for all y, x> y". In other words, there does NOT exist a largest integer.
 
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