Natural deduction system

$\{[p]_{(1)},q \}$
$\vdash q$
$\vdash_{(1)} (p \rightarrow q)$

$\{ p \rightarrow q , \lnot q \}$
$\vdash \{ p \rightarrow q , p \rightarrow \lnot q \}$
$\vdash \lnot p$

$p \land \lnot p\rightarrow p,p \land \lnot p \rightarrow \lnot p$
$\vdash \lnot(p \land \lnot p)$

$\lnot(p \land \lnot p)$
$\vdash \lnot p \lor \lnot \neg p$(De Morgan's Law)
$\vdash \lnot p \lor p$

$[p]_{(1)}$
$\vdash p \lor q$
$\vdash_{(1)} p \rightarrow (p\lor q)$

$[q]_{(2)}$
$\vdash p \lor q$
$\vdash_{(2)} q \rightarrow (p\lor q)$

$\{p \rightarrow (p \lor q) ,\lnot (p \lor q)\} \vdash \lnot p$(Modus tollens)
$\{q \rightarrow (p \lor q) ,\lnot (p \lor q)\} \vdash \lnot q$(Modus tollens)

$\lnot(p\lor q) \vdash (\lnot p \land \lnot q)$

$[p]_{(1)}$
$\vdash q$
$\vdash_{(1)} p\rightarrow q$

$[p]_{(2)}$
$\vdash \lnot q$
$\vdash_{(2)} p\rightarrow \lnot q$

$[p]_{(3)}$
$\vdash_{(3)} (p\rightarrow q)\land (p\rightarrow \lnot q)$
$\vdash \lnot p$

$(\lnot p \land \lnot q) \vdash \lnot p$

$\lnot p \vdash (p \rightarrow r)$

$(\lnot p \land \lnot q) \vdash \lnot q$

$\lnot q \vdash (q \rightarrow r)$

$\{[p \lor q]_{(1)},(\lnot p\land \lnot q)\}$
$\vdash \{(p\lor q),(p \rightarrow r),(q \rightarrow r) \}$
$\vdash r$(arbitrary)
$\vdash_{(1)} \lnot (p\lor q)$