Order statistics from exchangeable random variables are always sufficient

Let $(X_1,\ldots,X_n)$ be an exchangeable random vector with distribution function $F$, and denote by $Y_1\leq \cdots\leq Y_n$ the corresponding order statistics. We show that the conditional distribution of $(X_1,\ldots,X_n)$ given $(Y_1,\ldots,Y_n)$ does not depend on $F$.

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