rules-of-substitution

Similar to PL, FOL also have rules of substitution.

$$\begin{gathered} \text{displaylines}\neg (\forall xP(x))\equiv \exists x\neg (P(x))\dots DeMorga{n}^{\prime }sLaw \\ \neg (\exists xP(x))\equiv \forall x\neg (P(x))\dots DeMorga{n}^{\prime }sLaw \\ \forall x\forall y\equiv \forall y\forall x \\ \exists x\exists y\equiv \exists y\exists x \end{gathered}$$

Info

$\exists x\forall y\equiv \forall y\exists x$ is not valid.