\documentclass[convert={outfile=\jobname.svg}]{standalone}

\usepackage{proof}
\usepackage{amsmath}
\usepackage{utf8}

\begin{document}

$$
\infer[→~intro]{⊢ ¬(P∧Q) → ¬P∨¬Q}{
  \infer[∨~elim]{¬(P∧Q) ⊢ ¬P∨¬Q}{
    \infer[excl.~mid.]{\cdots ⊢ P∨¬P}{}
    &
    \infer[∨~elim]{\cdots, P ⊢ ¬P∨¬Q}{
      \infer[excl.~mid.]{\cdots ⊢ Q∨¬Q}{}
      &
      \infer{\cdots, Q ⊢ ¬P ∨ ¬Q}{\text{proved below}}
      &
      \infer[∨~intro]{\cdots ¬Q ⊢ ¬P∨¬Q}{
	\infer[assum]{\cdots, ¬Q ⊢ ¬Q}{} 
      }
    }
    &
    \infer[∨~intro]{\cdots, ¬P ⊢ ¬P∨¬Q}{
      \infer[assum]{\cdots, ¬P ⊢ ¬P}{} 
    }
  }
}
$$
\end{document}
