\begin{frame}[t]
  \frametitle{Quantifiers}

  \vspace{-1ex}
  Instead of `$x$ is green' we now want to say:
  \begin{itemize}
    \item `somebody is green', or
    \item `everybody is green'.
  \end{itemize}
  \pause
  
  \begin{goal}{Existential quantifier}
    \vspace{-1ex}
    \begin{talign}
      \myex{x}{\phi}  
    \end{talign}
    \vspace{-3ex}
    
    means: there exists some $x$ such that the formula $\phi$ is true.
  \end{goal}
  \pause

  \begin{exampleblock}{}
    \begin{malign}
      \myex{x}{P(x)} \quad\quad\hint{(somebody is green)}
    \end{malign}
  \end{exampleblock}
  \pause

  \begin{goal}{Universal quantifier}
    \vspace{-1ex}
    \begin{talign}
      \myall{x}{\phi} 
    \end{talign}
    \vspace{-3ex}
    
    means: for every $x$ the formula $\phi$ is true.
  \end{goal}
  \pause

  \begin{exampleblock}{}
    \begin{malign}
      \myall{x}{P(x)} \quad\quad\hint{(everybody is green)}
    \end{malign}
  \end{exampleblock}
  \vspace{10cm}
\end{frame}