De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 1/Reverse Implication

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Theorem

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


Proof

By the tableau method of natural deduction:

$\neg \paren {p \lor q} \vdash \neg p \land \neg q$
Line Pool Formula Rule Depends upon Notes
1 1 $\neg \paren {p \lor q}$ Premise (None)
5 5 $p$ Assumption (None)
3 2 $p \lor q$ Rule of Addition: $\lor \II_1$ 2
4 1, 3 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 3, 1
5 1 $\neg p$ Proof by Contradiction: $\neg \mathcal I$ 2 – 4 Assumption 2 has been discharged
6 6 $q$ Assumption (None)
7 6 $p \lor q$ Rule of Addition: $\lor \II_2$ 6
8 1, 7 $\bot$ Principle of Non-Contradiction: $\neg \mathcal E$ 7, 1
9 1 $\neg q$ Proof by Contradiction: $\neg \mathcal I$ 6 – 8 Assumption 6 has been discharged
10 1 $\neg p \land \neg q$ Rule of Conjunction: $\land \mathcal I$ 5, 9

$\blacksquare$