Be $G$ a connected graph with weights associated to its edges. Be $T(G)$ the graph that has the spanning trees of $G$ as vertex, and two spanning trees are adjacent to each other if and only if each one of them has only one edge that the other doesn't have. To each vertex of $T(G)$ we associate the weight of the spanning tree that correspond to $G$.
Be $T_1$ any vertex of $T(G)$, that is, any spanning tree of $G$. Proof that $T_1$ is a local minimum if and only if $T_1$ is global minimum.
To put it in another words, proof that $T_1$ has weight minor or equal than all its neighbours in $T(G)$ if and only if $T_1$ is an minimum spanning tree.