# How to derive ~ ( P&Q) from ~ P using natural deduction?

Certainly if P is false, (P&Q) cannot be true.

But how to prove this using natural deduction?

I'd propose as a direct proof the following derivation :

(1) ~P ( Premise )

(2) ~P v ~Q ( v - intro)

(3) ~ ( P & Q) ( DeMorgan)

1) $$\lnot P$$ --- premise
2) $$(P \land Q)$$ --- assumed [a]
3) $$P$$ --- from 2) by $$(\land \text E)$$
4) $$\bot$$ --- from 1) and 2), by $$(\lnot \text E)$$ (alternatively, using $$(\to \text E)$$, if $$\lnot P$$ is defined as $$P \to \bot$$)
5) $$\lnot (P \land Q)$$ --- from 2) and 4) by $$(\lnot \text I)$$, discharging [a].