Assuming $x$ does not occur free in $A$, prove that
$$(\exists x (A \to B)) \leftrightarrow (A \to ( \exists x B))$$
using any of the following axioms; MP, HS, or the Deduction Theorem.
1) $A \to (B \to A)$
2) $(A \to (B \to C)) \to ((A \to B) \to (A \to C))$
3) $(\neg A \to \neg B) \to (B \to A)$
4) $(\forall x A) \to A$
5) $(\forall x (A \to B)) \to (A \to \forall x B)$
First of all, I don't know how to convert the existential quantifier into the universal quantifier.
Is $\exists x A$ the same as $\neg (\forall x) \neg A$?
Is $\neg(\forall x) \neg A$ the same as $A$?
Second, I'd appreciate your help with the original question.