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\begin{frame}
  \frametitle{Modal Logic Equivalence}
  
  \begin{goal}{We define \;\;$ \phi \equiv \psi $\;\; as} 
    In every world $w$ in every Kripke model $\mathcal{M}$ 
    \begin{talign}
      \mathcal{M},w \models \phi  \;\riff \; \mathcal{M},w \models \psi
    \end{talign}
  \end{goal}
  \pause
  
  \begin{block}{Alternative definition of modal equivalence}
    \begin{malign}
      \phi \equiv \psi  \;\;\;\;\riff \;\;\; \;
      \phi\models\psi  \;\;\mbox{ \aemph{and} } \;\;  \psi \models \phi
    \end{malign}
  \end{block}
  \pause
  
  \begin{exampleblock}{}
  \vspace{-3ex}
  \begin{eqnarray*}
    \mpause[1]{ \all \phi &  \equiv  &   \neg \some \neg \phi }  \\
    \mpause{ \some \phi &  \equiv  &   \neg \all \neg \phi } \\ 
    \mpause{ \some \neg \phi & \equiv  &   \neg \all \phi } \\  
    \mpause{ \all( \phi \wedge\psi) &  \equiv  &   \all \phi  \wedge \all \psi } \\ 
    \mpause{ \all( \phi \vee\psi) &  \not\equiv  &   \all \phi  \vee \all \psi } \\ 
    \mpause{ \phi \vee \psi & \equiv & \neg \phi \to\psi \;\;\mpause{^{\hint{$\star$}}}}
  \end{eqnarray*}
  \vspace{-3ex}
  
  \mpause[7]{\hint{$^{\hint{$\star$}}$: all equivalences from propositional logic hold also modal !}}
  \end{exampleblock}
\end{frame}