The labeled multigraphs of treewidth at most two can be described using a simple term language over which isomorphism of the denoted graphs can be finitely axiomatized. We formally verify soundness and completeness of such an axiomatization using Coq and the mathematical components library. The completeness proof is based on a normalizing and confluent rewrite system on term-labeled graphs. While for most of the development a dependently typed representation of graphs based on finite types of vertices and edges is most convenient, we switch to a graph representation employing a fixed type of vertices shared among all graphs for establishing confluence of the rewrite system. The completeness result is then obtained by transferring confluence from the fixed-type setting to the dependently typed setting.