# Can an if and only if condition be stated alternatively?

Can

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

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

i.e.

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

• What do you mean by $::$? I haven't seen that notation before. Oct 5 '12 at 4:04
• @AlexanderGruber: Removed it, it wasn't relevant to the question! Oct 5 '12 at 4:13

$P \Leftrightarrow Q$ 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: $\neg P \Leftrightarrow \neg Q$