\begin{frame} \frametitle{Proof by Contradiction} Assume that you have derived \begin{align*} \parbox{.8cm}{\centerline{$\neg\alpha$}\centerline{$\vdots$}\centerline{$\bot$}} \end{align*} \pause Then also \begin{align*} \parbox{3cm}{ \framebox{\parbox{.8cm}{\centerline{$\neg\alpha$}\centerline{$\vdots$}\centerline{$\bot$}}}\\[1ex] $\neg\neg \alpha$ \quad\quad $\neg_i$\\[.5ex] $\alpha\hphantom{\neg\neg}$ \quad\quad $\neg\neg_e$ } \end{align*} \pause This is known as \emph{Proof by Contradiction (PBC)}! \pause\smallskip Also known as \emph{Reductio ad Absurdum (RAA)}! \end{frame}