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$P \Leftrightarrow (\exists x: A)$ be stated alternatively as

$\overline{P} \Leftrightarrow \neg (\exists x: A)$


$\overline{P} \Leftrightarrow (\forall x: \neg A)$ ?

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What do you mean by $::$? I haven't seen that notation before. – Alexander Gruber Oct 5 '12 at 4:04
@AlexanderGruber: Removed it, it wasn't relevant to the question! – TCSGrad Oct 5 '12 at 4:13
up vote 4 down vote accepted

$P \Leftrightarrow Q$ is the same as $P \Rightarrow Q$ and $Q \Rightarrow P$
Those may be converted as $\neg Q \Rightarrow \neg P$ and $ \neg P \Rightarrow \neg Q$
Which is then indeed what you are asking about: $\neg P \Leftrightarrow \neg Q$

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