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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}