I'm trying to establish a Fitch proof regarding this question but I keep getting stuck.
$$\forall x \forall y \; Rxy \quad\vdash_F\quad \forall y \forall x \; Ryx$$
To be honest I have no idea how to get from the premise to the conclusion. I can't figure out to switch the constants' places. Do I have to show that $x=y$?
Hint: $~$ You can show that $~~\forall x~Px \vdash_{\small F} \forall y~Py~~$ by:
$\def\fitch#1#2{~~~~\begin{array}{|l}#1\\\hline#2\end{array}}\fitch{~~1.~\forall x~Px}{\fitch{~~2.~\boxed a}{~~3.~Pa\quad\forall E~1}\\~~4.~\forall y~Py\quad\forall I~2{-}3}$
So do something similar...