\begin{frame}
  \frametitle{Not Forall and Not Exists}

  Let us reconsider two examples
  \begin{align*}
    \neg\myall{x}{P(x)} &\quad\quad \hint{Not everybody is green.}\\
    \myex{x}{\neg P(x)} &\quad\quad \hint{There exists someone who is not green.}
  \end{align*}
  \pause
  \emph{Note that both statements are equivalent!}
  \pause
  \bigskip
  
  \begin{goal}{}
    In general we have the following equivalences:
    \begin{talign}
      \neg\myall{x}{\phi} \quad\iff\quad \myex{x}{\neg \phi} \\[1ex]
      \neg\myex{x}{\phi} \quad\iff\quad \myall{x}{\neg \phi} \\[1ex]
      \myall{x}{\phi} \quad\iff\quad \neg\myex{x}{\neg \phi} \\[1ex]
      \myex{x}{\phi} \quad\iff\quad \neg\myall{x}{\neg \phi}
    \end{talign}
  \end{goal}
  \pause
  Note that all these equivalences follow from each other.
\end{frame}