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

Theorem

$\paren {\neg \paren {p \lor q} } \implies \paren {\neg p \land \neg q}$

Proof

By the tableau method of natural deduction:

$\vdash \paren {\neg \paren {p \lor q} } \implies \paren {\neg p \land \neg q} $

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

$\blacksquare$