\documentclass{article} 
\usepackage{proof}
\newcommand{\limp}{\supset}
\renewcommand{\lnot}{\neg}
\renewcommand{\land}{\wedge}
\renewcommand{\lor}{\vee}
\newcommand{\A}{\mathbf{A}}
\newcommand{\B}{\mathbf{B}}
\newcommand{\C}{\mathbf{C}}

\begin{document} 
$$
\infer[\limp-I^u]{(\lnot\A\lor\lnot\B)\limp\lnot(\A\land\B)}
 {\infer[\limp-I^v]{\lnot(\A\land\B)}{\infer[\lor-E^{\{w,x\}}]{\bot}
 {\infer[u]{\lnot\A\lor\lnot\B}{} & \infer[\limp-E]{\bot}
   {\infer[w]{\lnot\A}{} & \infer[\land-E_L]{\A}{\infer[v]{\A\land\B}{}}} &
  \infer[\limp-E]{\bot}
   {\infer[x]{\lnot\B}{} & \infer[\land-E_R]{\B}{\infer[v]{\A\land\B}{}}}}}}}
$$
\end{document}