Flip and negate

$\lnot \forall x P(x) \equiv \exists x \lnot P(x)$. Not everything has a property exactly when something lacks it.

$\lnot \exists x P(x) \equiv \forall x \lnot P(x)$. Nothing has a property exactly when everything lacks it.

For nested quantifiers, move from the outside in: flip each quantifier, then negate the statement at the end.