According to the book Topological Graph Theory by Gross and Tucker, given a cellular embedding of a graph on a surface (by 'surface' I mean here a sphere with $n\geq 0$ handles), one can define a dual multigraph by treating the faces of the original graph embedding as vertices and adding an edge between two vertices for every edge the corresponding faces have in common in the original graph.
My question is this: Let $G$ be cellularly embedded in some surface and let $G'$ be the dual of this embedding. If $\delta (G)\geq 3$, can I be sure that $G'$ is simple (there is at most one edge between two vertices)?
This should be simple, but I am not comfortable enough with proofs in topological graph theory.
Note: $\delta (G)$ denotes the minimum degree of vertices in $G$.
Edit: Follow-up to this question: When is the dual graph simple?