Quasicrystals are interesting objects, with their diffraction patterns consisting of pure bright peaks but showing certain symmetries which are forbidden in crystals. In mathematics, diffraction consisting of pure bright peaks is called pure point diffraction.
In the study of pure point diffraction it has been known that a regular model set is pure point diffractive, but the converse has been a conjecture. We prove that the converse is also true for substitution point sets in R^d without the condition of regularity. We provide the equivalence between model sets and pure point diffractive sets introducing a new type of coincidence. The result is fairly general, so that it covers Pisot substitution point sets.