Minimal models and boundedness of stable varieties

We consider a class of stable smoothable n-dimensional varieties, the analogs of stable curves. Assuming the minimal model program in dimension n+1, we prove that this class is bounded. From Kollar's method of constructing projective moduli spaces we get as a corollary that minimal model program in dimension n+1 implies the existence of a projective coarse moduli space for stable smoothable n-folds.