Regras Derivadas

Leis de De Morgan

$$\begin{array}{rlr}& \neg \forall xp⊣⊢\exists x\neg p& \neg \exists xp⊣⊢\forall x\neg p\end{array}$$

Regras Gerais

$$\begin{array}{rlr}& \left(\forall xp\right)\wedge \left(\forall xq\right)⊣⊢\forall x\left(p\wedge q\right)& \forall x\forall yp⊣⊢\forall y\forall xp\\ & \left(\exists xp\right)\vee \left(\exists xq\right)⊣⊢\exists x\left(p\vee q\right)& \exists x\exists yp⊣⊢\exists y\exists xp\end{array}$$

Se $x$ não ocorre em $q$: $$\begin{array}{rlr}& q\wedge \forall xp⊣⊢\forall x\left(q\wedge p\right)& q\vee \forall xp⊣⊢\forall x\left(q\vee p\right)\\ & q\wedge \exists xp⊣⊢\exists x\left(q\wedge p\right)& q\vee \exists xp⊣⊢\exists x\left(q\vee p\right)\\ & q\to \forall xp⊣⊢\forall x\left(q\to p\right)& q\to \exists xp⊣⊢\exists x\left(q\to p\right)\\ & \left(\forall xp\right)\to q⊣⊢\exists x\left(p\to q\right)& \left(\exists xp\right)\to q⊣⊢\forall x\left(p\to q\right)\end{array}$$