Regras Derivadas

Leis de De Morgan

$$ \begin{aligned} & \neg\forall x~ p\ \bproof \exists x~ \neg p \cr & \neg\exists x~ p \bproof \forall x~ \neg p \end{aligned} $$

Regras Gerais

$$ \begin{aligned} & \del{\forall x~ p} \land \del{\forall x~ q} \ \bproof\ \forall x~\del{p \land q} & \forall x~\forall y\ p \ \bproof\ \forall y~\forall x~ p \cr & \del{\exists x\ p} \lor \del{\exists x\ q} \bproof \exists x\ \del{p \lor q} & \exists x\ \exists y\ p \bproof \exists y \exists x ~ p \end{aligned} $$

Se $x$ não ocorre em $q$: $$ \begin{aligned} & q \land \forall x~ p \bproof \forall x\del{q \land p} &q \lor \forall x~ p \bproof \forall x\del{q \lor p} \cr & q \land \exists x~ p \bproof \exists x\del{q \land p} & q \lor \exists x~ p \bproof \exists x\del{q \lor p} \cr & q \to \forall x~ p \bproof \forall x\del{q \to p} & q \to \exists x~ p \bproof \exists x\del{q \to p} \cr & \del{\forall x~ p} \to q \bproof \exists x\del{p \to q} & \del{\exists x~ p} \to q \bproof \forall x\del{p \to q} \end{aligned} $$