We study a general social choice environment that has multiple agents, a finite set of alternatives, and independent and diffuse information. We show that for any Bayesian incentive compatible mechanism, there exists a deterministic mechanism that i) is Bayesian incentive compatible; ii) delivers the same interim expected utilities/allocation probabilities for all the agents; and iii) delivers the same ex ante expected welfare. Our result holds in settings with a rich class of utility functions, multi-dimensional types, interdependent valuations, and non-transferable utilities. More importantly, the result recovers the optimality of deterministic mechanisms (whether in terms of revenue or efficiency), which sharply contrasts with the existing results in the screening literature. To prove our result, we develop a new methodology of “mutual purification”, and establish its link with the literature of mechanism design.