De Morgan's Laws (Logic)/Conjunction/Formulation 1/Forward Implication

Theorem

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

Proof

 * align="right" | 6 ||
 * align="right" | 1, 5
 * $\bot$
 * $\neg \mathcal E$
 * 2, 5
 * 2, 5


 * align="right" | 8 ||
 * align="right" | 1, 7
 * $\bot$
 * $\neg \mathcal E$
 * 3, 7
 * align="right" | 9 ||
 * align="right" | 1, 4
 * $\bot$
 * $\lor \mathcal E$
 * 4, 5-6, 7-8
 * align="right" | 10 ||
 * align="right" | 1
 * $\neg \left({\neg p \lor \neg q}\right)$
 * Proof by Contradiction
 * 4, 9
 * align="right" | 10 ||
 * align="right" | 1
 * $\neg \left({\neg p \lor \neg q}\right)$
 * Proof by Contradiction
 * 4, 9