\begin{frame}{Modification of Environments}
  In order to determine the truth values for \alert{$\forall x$}- en \alert{$\exists x$}-formulas\\
  we must be able to modify the interpretation of the variable \alert{$x$}.
  
  \begin{definition}[Modified Environment]
    Let $\saluf \funin \textbf{var} \to \model{\adomain}$ be an environment,
    and $\avarsynvar$ a variable.
    \smallskip
    
    By $\saluf[\avarsynvar\mapsto \forestgreen{a}]$ we denote
    the environment that interprets $\avarsynvar$ as $\forestgreen{a}$,
    and that for all other variables acts in the same way as $\saluf$:
    \begin{talign}
      \fap{\saluf[\avarsynvar\mapsto \forestgreen{a}]}{\bvarsynvar} =
      \begin{cases}
        \forestgreen{a} & \text{if } \bvarsynvar = \avarsynvar
        \\
        \aluf{\bvarsynvar}
          & \text{if } \bvarsynvar \neq \avarsynvar
      \end{cases}
    \end{talign}
  \end{definition}\pause{}
  
  \begin{exampleblock}{}
    More modifications are possible.
    \smallskip
    
    For  $\salufacc = \saluf[x\mapsto \forestgreen{a}][z\mapsto \forestgreen{b}]$ we have:
    \begin{talign}
      \alufacc{\bvarsynvar} = \fap{\saluf[x\mapsto \forestgreen{a}][z\mapsto \forestgreen{b}]}{\bvarsynvar} =
      \begin{cases}
        \forestgreen{a} & \text{if } \bvarsynvar = x \\
        \forestgreen{b} & \text{if } \bvarsynvar = z \\
        \aluf{\bvarsynvar} & \text{if } \bvarsynvar \neq x \text{ and } \bvarsynvar \neq z 
      \end{cases}
    \end{talign}
  \end{exampleblock}
\end{frame}