[tex] \forall x P(x) \Leftrightarrow \neg \exists x \neg P(x) [/tex]
Which is why you can't prove it.
To prove it, something must first exist in the universe of discourse, call it "a"
[tex] a\ exists \bigwedge \forall x P(x) \Rightarrow \exists x ¬P(x) [/tex]
Your proposition is invalid for all models based on the empty set.
Or, in other words, your proposition is valid for all universes of discourse except the one universe in which nothing exists.
Logic is spooky....