Im not quite sure what you mean with your last question. But this is how i understand the sentence.
"There exist an X where all Y is in the same function p(X,Y) which implies that for all Y is there an X in the same function p(X,Y)"
First of all i need to remove the imply arrow by negating the right side and split up the left and right part like this:
You can then remove by introducing a new constant for X.