\documentclass{article}
\usepackage[numbers]{natbib}
\usepackage{apxproof}
\begin{filecontents*}[overwrite]{refs.bib}
@article{thomas1998update,
title={An update on the four-color theorem},
author={Thomas, Robin},
journal={Notices of the AMS},
volume={45},
number={7},
pages={848--859},
year={1998}
}
@article{robertson1986graph,
title={Graph minors. II. Algorithmic aspects of tree-width},
author={Robertson, Neil and Seymour, Paul D.},
journal={Journal of algorithms},
volume={7},
number={3},
pages={309--322},
year={1986},
publisher={Elsevier}
}
\end{filecontents*}
\begin{document}
Here is a reference that is not in appendix \cite{thomas1998update}.
\begin{toappendix}
The same reference should be displayed here \cite{thomas1998update} since it's already been used.
However, the following won't \cite{robertson1986graph}.
\end{toappendix}
\bibliographystyle{plainnat}
\bibliography{refs}
\end{document}