Superdeduction is a focusing method especially designed to ease the use of first-order theories in predicate logic. Superdeduction modulo, which results of the combination of deduction modulo and superdeduction, enables then the user to make a distinct use of computational and reasoning axioms. Although soundness and completeness of superdeduction modulo are squarely ensured, it is not the case of cut-free superdeduction modulo since both deduction modulo and superdeduction may break cut-elimination. In this paper we prove the soundness and completeness of cut-free superdeduction modulo: we show that by using permutations of inference rules, any cut-free proof in deduction modulo can be transformed into a cut-free proof in superdeduction modulo and conversely. As a corollary we obtain that cut-elimination for deduction modulo (which is well-studied) is equivalent to cut-elimination for superdeduction modulo provided that some hypotheses on the synchrony of reasoning axioms are verified. Finally we propose a tableau method based on superdeduction modulo which is sound and complete provided that cut-elimination holds.