\begin{frame}{Alternative Translations}
  We can translate the sentence:
  \begin{equation*}
    \sentence{Nobody is perfect.}
  \end{equation*}
  in two ways:
  \begin{talign}
    \mpause[1]{\sentence{There does not exist anybody who is perfect.} & :\;\;}
    \mpause{\formula{\lognot{\existsst{x}{\,\unpred{P}{x}}}}}
    \\
    \mpause{\sentence{Everybody is not perfect.} & :\;\;}   
    \mpause{\formula{\forallst{x}{\,\lognot{\unpred{P}{x}}}}}
  \end{talign}
  \updatepause
  These three English sentences have the same meaning.
  \bigskip
  
  The two formulas are \emph{logically equivalent}:
  \pause
  \begin{talign}
    \formula{\lognot{\existsst{x}{\,\unpred{P}{x}}}} 
    \logequiv
    \formula{\forallst{x}{\,\lognot{\unpred{P}{x}}}}   
  \end{talign}
  \pause{}
  It holds in general, for every formula $\aform\,$: 
  \begin{talign}
    \formula{\lognot{\existsst{x}{\,\aform}}} 
    \logequiv
    \formula{\forallst{x}{\,\lognot{\aform}}}
  \end{talign}   
\end{frame}