\subsubsection{Quantifiers}
\paragraph{Universal quantification}
Additional rules are needed for the universal quantifier. * side condition is that $x$ is not free in any assumption of $\Gamma$
\begin{align*}
    \begin{prooftree}
        \hypo{\Gamma \vdash A}
        \infer1[$\forall$-I*]{\Gamma \vdash \forall x.A}
    \end{prooftree}
    \qquad
    \begin{prooftree}
        \hypo{\Gamma \vdash \forall x.A}
        \infer1[$\forall$-E]{\Gamma \vdash A[x \rightarrow t]}
    \end{prooftree}
\end{align*}
Again here be mindful not to capture free variables.

\paragraph{Existential quantification}
Additional rules are needed for the existential quantifier. ** side condition is that $x$ is neither free in B nor free in $\Gamma$
\begin{align*}
    \begin{prooftree}
        \hypo{\Gamma \vdash A[x \mapsto t]}
        \infer1[$\exists$-I]{\Gamma \vdash \exists x.A}
    \end{prooftree}
    \qquad
    \begin{prooftree}
        \hypo{\Gamma \vdash \exists x.A}
        \hypo{\Gamma, A \vdash B}
        \infer2[$\exists$-E**]{\Gamma \vdash A[x \rightarrow t]}
    \end{prooftree}
\end{align*}
Again here be mindful not to capture free variables.
