Prove (¬(∀(¬()))) ⊢ (∃ ()) by Natural Deduction I want to prove (¬(∀(¬()))) ⊢ (∃()) using only the basic rules of the Natural Deduction system for propositional logic and predicate logic. 
I am not sure how to get rid of the negation before the universal quantifier. 
How should I prove this? 

If in doubt, try reductio ....
So after the initial premiss, assume $\neg\exists xPx$
Now what?
You'll have to make another assumption to get anywhere ....
So suppose $Pa$ (the obvious thing ...why??)
Then you can infer $\exists xPx$, contradiction!
So you can infer $\neg Pa$
And that gives you $\forall x\neg Px$
And now the end is in sight, because this contradicts the initial premiss ...
Join up the dots, and finish the proof.
