\setlength\subproofhorizspace{1.3em}
\begin{logicproof}{2}
  \exists x \; \lnot P(x)           & \prem\\
  \begin{subproof}
    \forall x \; P(x)		        & $\assum$\\
    P(x_0)                          & $\foralle2$\\
    \begin{subproof}
        \lnot P(x_0)                & $\assum$ $\freshVar{$x_0$}$\\
        \bot                        & $\nege3,4$
    \end{subproof}
    \bot                            & $\existe1,3-5$
  \end{subproof}
  \lnot \forall x \; P(x)           & $\negi2-5$
\end{logicproof}