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Relation to existential quantification

Equivalent to the existential quantifier under negation by

\[\begin{align*}
  \forall x \> \lnot P(x) \> & \equiv \> \lnot \exists x \> P(x) \\
  \lnot \forall x \> P(x) \> & \equiv \> \exists x \> \lnot P(x),
\end{align*}
\]

which is akin De Morgan’s laws, with \(\forall\) generalizing conjunction.