Summary of quantifiers and negation

Definitions.
- $\forall x,\ P(x)$: $P$ holds for all $x$.
- $\exists x,\ P(x)$: some $x$ satisfies $P$.
- $\exists! x,\ P(x)$: exactly one $x$ satisfies $P$.

Negation.
- $\neg\forall x,\ P(x)\equiv\exists x,\ \neg P(x)$.
- $\neg\exists x,\ P(x)\equiv\forall x,\ \neg P(x)$.
- $\neg\exists!x,\ P(x)$: no $x$ or at least two $x$ satisfy $P$.

Order. $\forall\exists$ does not imply $\exists\forall$ in general.

Vacuous truth. $\forall x\in\emptyset,\ P(x)$ is true.

Conclusion. Quantifiers are the backbone of mathematical statements; negation and order are critical for correct reasoning.