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In this article, we show that the extension of the resolution proof system to deduction modulo is equivalent to the cut-free fragment of the sequent calculus modulo. The result is obtained through a syntactic translation, without using any cut-elimination procedure. Additionally, we show Skolem theorem and inversion/focusing results. Thanks to the expressiveness of deduction modulo, all these results also apply to the cases of higher-order resolution, Peano's arithmetic and Gentzen's LK.