\begin{frame}
  \frametitle{MT and $\neg\neg_i$ as ``derived rules''}
  
  \begin{goal}{}
    The $\neg\neg_i$ rule derives \;$\neg\neg \alpha$\; from \;$\alpha$.
    \medskip\pause
    
    We can derive it using other rules as follows:
    \smallskip
    \begin{tikzpicture}
    \naturaldeduction{
      \mpause[1]{
        \proofstep{$\alpha$}{premise};
      }
      \mpause{
        \proofbox{
          \mpause{
            \proofstep{$\neg \alpha$}{assumption};
          }
          \mpause{
            \proofstep{$\bot$}{$\neg_e$ 1,2};
          }
        }
      }
      \mpause{
        \proofstep{$\neg\neg \alpha$}{$\neg_i$ 2--3};
      }
    }
    \end{tikzpicture}
    \pause\pause\pause\pause\pause\pause
    \smallskip
    
    Thus: $\alpha \;\vdash \neg\neg\alpha$\;.
  \end{goal}
\end{frame}