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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.

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

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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 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 ...

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