Proving a Logic Relation?

Question

How i can prove this term ( x,y are logic variables and p(x,y) is any logic function):

(Exist X)(For All Y) P(X,Y) => (For All Y)(Exist X) P(X,Y)

this is clear for me that this is true, but how i can prove it?

P(X,Y) is a combination of x , y , "logic and", "logic or", and other logic operators.
So, my question is how to prove that the first part of term exigency the second part. means:

Why when (Exist X)(For All Y) P(X,Y)
we have (For All Y)(Exist X) P(X,Y)

it is related to Logic mathematic.

Please give me a proving!

Answer 1

Let's assume there's Y for each an X such that P(X,Y) does not exist.

Now there is x1 that for all Y P(X,Y), but for the Y we supposed that exists such an X does not exist, Particularly, x1 is not such an X.

Answer 2

Without using proof by contradiction, this is what we can do.

Assume LHS. Then there is an X such that P(X,Y) holds for all Y. Fix a choice of X for this - call it X0. Now to prove the RHS, given any Y, we have P(X0,Y) by our definition of X0. Hence, P(X,Y) holds for some X.