\begin{frame}
   \frametitle{Truth of Boxes: $\all \phi$ }
  
  \begin{goal}{$w \fc \all \phi$}
    The formula \;\;$\all \phi$ is true in world $w$\;\;
    if $\phi$ is true in all worlds $w'$ with $R(w,w')$.
  \end{goal}
  \pause
  
  \begin{block}{}
  As a formula:\quad
    $w \fc \all \phi \quad\DARKRED{\iff}\quad
    \myall{w'}{\big( R(w,w') \;\to\; w' \fc \phi \big)}$
  \end{block}
  \pause
  
  \begin{exampleblock}{}
    \exampleA
    
    For example,
    \begin{talign}
      \mpause[1]{ w_1  &\mpause{\notfc}  \all\lab{p} } &
      \mpause{ w_3  &\mpause{\fc}         \all\lab{q} } &
      \mpause{ w_1  &\mpause{\fc}         \all\lab{(q \vee r)} } &
      \mpause{ w_2  &\mpause{\fc}         \all\lab{\bot} }
    \end{talign}
  \end{exampleblock}
  \pause\pause\pause\pause\pause\pause\pause\pause\pause
  \hint{Note that $\all\lab{\bot}$ holds only in worlds without outgoing edges!}
\end{frame}