De Morgan's Laws (Logic)/Disjunction of Negations/Formulation 1/Forward Implication

Theorem

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

$\neg p \lor \neg q \vdash \neg \paren {p \land q} $
Line	Pool	Formula	Rule	Depends upon	Notes
1	1	$\neg p \lor \neg q$	Premise	(None)
2	2	$p \land q$	Assumption	(None)
3	2	$p$	Rule of Simplification: $\land \EE_1$	2
4	2	$q$	Rule of Simplification: $\land \EE_2$	2
5	5	$\neg p$	Assumption	(None)
6	2, 5	$\bot$	Principle of Non-Contradiction: $\neg \EE$	3, 5
7	7	$\neg q$	Assumption	(None)
8	2, 7	$\bot$	Principle of Non-Contradiction: $\neg \EE$	4, 7
9	1, 2	$\bot$	Proof by Cases: $\text{PBC}$	1, 5 – 6, 7 – 8	Assumptions 5 and 7 have been discharged
10	1	$\neg \paren {p \land q}$	Proof by Contradiction: $\neg \II$	2 – 9	Assumption 2 has been discharged

$\blacksquare$