∀x∀y(R(x,y)∧R(y,x))⇒∃zR(z,z)
https://scrapbox.io/files/65f957b253380b0025e2bb50.svg
code:proof.tikz(tex)
\usepackage{fitch}
\begin{document}
$\Large\begin{nd}
\hypo {h} {\forall x\forall y(R(x,y)\land R(y,x))}
\have {u} {\forall y(R(u,y)\land R(y,u))} \Ae{h}
\have {uu} {R(u,u)\land R(u,u)} \Ae{u}
\have {ru} {R(u,u)} \ae{uu}
\have {er} {\exists z R(z,z)} \Ei{ru}
\close
\have {ae} {\forall u\exists zR(z,z)} \Ai{er}
\have {e} {\exists zR(z,z)} \Ae{ae}
\end{nd}$
\end{document}