\documentclass[a4paper,12pt]{scrartcl}
\usepackage[utf8x]{inputenc}

%opening
\title{AR22}
\author{Malvin Gattinger}
\date{}

\begin{document}

\newcommand{\so}{\rightarrow}

\parindent0cm

\section*{AR22}

\begin{center}
$ \lnot ( p \so q ) \dashv \vdash p \land \lnot q $
\end{center}

$\vdash$ : \\
$\begin{array}{llll}
1    & (1) & \lnot (p \so q) & A \\
2    & (2) & \lnot ( p \land \lnot q ) & A \\
3    & (3) & p & A\\
4    & (4) & \lnot q & A \\
3,4  & (5) & p \land \lnot q & 3,4 \land I \\
2,3  & (6) & q & 2, 5 RAA (4) \\
2    & (7) & p \so q & 6 \so I (3) \\
1    & (8) & p \land \lnot q & 1,7 RAA (2) \\
\end{array}$

\bigskip

$\dashv$ : \\
$\begin{array}{llll}
1   & (1) & p \land \lnot q  & A \\
2   & (2) & p \so q          & A \\
1   & (3) & p                & 1 \land E \\
1   & (4) & \lnot q          & 1 \land E \\
1,2 & (5) & q                & 2,3 \so E \\
1   & (6) & \lnot (p \so q)  & 4,5 RAA (2) \\


\end{array}$

\end{document}