Suppose the heuristic function is h(s) = 0.

Then the priority queue key will be h(s) + d(s) = d(s).

If the children of a vertex at depth k are added to the priority queue they will go to the end (since they are at depth k + 1).

Therefore all vertices at depth k will be examined before vertices at depth k + 1 — the definition of breadth-first search.