# Stuck on proving demorgans with quantifiers [closed]

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.

-

## closed as unclear what you're asking by 900 sit-ups a day, mookid, Rick Decker, hardmath, Michael AlbaneseJul 16 at 0:05

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

Please reveal which quantifier rules you're allowed to use here. "UG box" and "RAA box" don't seem to be standard notation; they might be specific to the text you're working from. –  Henning Makholm Oct 9 '12 at 18:47

## 1 Answer

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 ...

-