Stuck on proving demorgans with quantifiers

Question

What I'm trying to prove, using propositional and quantifier rules, is

$$\neg \exists{x} \; A(x) \iff \forall{x} \; \neg A(x).$$

So far, I've only started proving it left to the right, and I'm stuck. What I have:

1: $\neg \exists{x} \; A(x)$

2: x fresh 2. open UG box

3: $\neg\neg A(x)$ 3. open RAA box

n-2. # n-2. close RAA

n-1. not A n-1. RAA 3 to n-2, close UG box

n. $\forall{x} \; \neg A(x)$

I just don't know how to deal with the 'not' in line 1.

Answer 1

Hint. Try this strategy instead. After line 1, start a subproof with the assumption $A(a)$. That quickly yields a contradiction (why)? So you can use RAA to conclude $\neg A(a)$. And now the end is in sight ...