De Morgan's Laws (Logic)/Conjunction of Negations/Formulation 2/Proof 1
Jump to navigation
Jump to search
Theorem
- $\vdash \paren {\neg p \land \neg q} \iff \paren {\neg \paren {p \lor q} }$
Proof
By the tableau method of natural deduction:
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): Disjunction of Negations: Formulation 1 | |
3 | $\left({\neg p \land \neg q}\right) \implies \left({\neg \left({p \lor q}\right)}\right)$ | Rule of Implication: $\implies \II$ | 1 – 2 | Assumption 1 has been discharged | ||
4 | 4 | $\neg \left({p \lor q}\right)$ | Assumption | (None) | ||
5 | 4 | $\neg p \land \neg q$ | Sequent Introduction | 4 | De Morgan's Laws (Logic): Disjunction of Negations: Formulation 1 | |
6 | $\left({\neg \left({p \lor q}\right)}\right) \implies \left({\neg p \land \neg q}\right)$ | Rule of Implication: $\implies \II$ | 4 – 5 | Assumption 4 has been discharged | ||
7 | $\left({\neg p \land \neg q}\right) \iff \left({\neg \left({p \lor q}\right)}\right)$ | Biconditional Introduction: $\iff \II$ | 3, 6 |
$\blacksquare$