$$\forall x ¬(\forall y P(x, y)) \rightarrow \forall x \forall y ¬P(x, y)$$
I'm trying to intuitively understand this idea by thinking about it in terms of English. The second half is easy. Where P is "x likes y" and x is the set of girls and y is the set of guys, then we can say it's not the case that every girl likes every guy.
The first half is more difficult to say. Where could I even begin to translate that? It's not the case that not every guy is liked by a girl? This is assuming the negation is distributed to Ay and P(x,y).
This isn't homework by the way, and I already know this is a false proposition. I just want to get the intuition.