\documentclass{article}
\input nd
\begin{document}
% The proof of some hypothesis, laid out in a style that I like.  Note the
% consistent naming scheme for inference rules: `ndimpi' is implication
% introduction, `ndnote' is negation (not) elimination, `ndexistsi' is
% existential introduction, etc.
$$
\ndimpi
  {\ndexistse
    {\ndlabel{\vdash \ndexists x. A}{u}}
    {\ndimpe
      {\ndforalle
         {\vdash \ndforall x. (A\ndimp B)}
         {\vdash ([a/x]A)\ndimp B}}
      {\ndlabel{\vdash [a/x]A}{w}}
      {\vdash B}}
    {\vdash B}
    {a, w}}
  {\vdash (\ndexists x. A) \ndimp B}
  {u}
$$
\end{document}
