De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 2/Forward Implication

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Theorem

$\left({\neg p \land \neg q}\right) \implies \left({\neg \left({p \lor q}\right)}\right)$


Proof

By the tableau method of natural deduction:

$\left({\neg p \land \neg q}\right) \implies \left({\neg \left({p \lor q}\right)}\right)$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg p \land \neg q$ Assumption (None)
2 1 $\neg \left({p \lor q}\right)$ Sequent Introduction 1 De Morgan's Laws (Logic): Conjunction of Negations: Formulation 1
3 $\left({\neg p \land \neg q}\right) \implies \left({\neg \left({p \lor q}\right)}\right)$ Rule of Implication: $\implies \mathcal I$ 1 – 2 Assumption 1 has been discharged

$\blacksquare$


Sources