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\begin{frame}
  \frametitle{Completeness Theorem}

  \begin{goal}{Completeness}
    \vspace{-1ex}
    \begin{talign}
      \phi_1,\ldots,\phi_n \;\models \; \psi 
      \quad\Rightarrow\quad 
      \phi_1,\ldots,\phi_n \;\vdash \; \psi 
    \end{talign}
  \end{goal}
  \pause\bigskip

  \begin{block}{Explanation}
    \pause
    In a slogan: everything true is derivable.
    \bigskip\pause
    
    If $\psi$ follows semantically from $\phi_1,\ldots,\phi_n$,\\
    then $\psi$ can be derived syntactically from premises $\phi_1,\ldots,\phi_n$.
    \bigskip\pause

    This means that the syntactic derivation rules are strong enough
    to derive every semantic conclusion.
    \bigskip\pause
    
    
    Thus the system is \emph{complete}: no more rules are needed.
  \end{block}
\end{frame}