This paper presents a highly configurable deduction system for predicate logic called superdeduction modulo, resulting of the combination of deduction modulo and superdeduction which are two orthogonal presentations of predicate logic. It enables the user to make a distinct use of computational and reasoning axioms in a sound and complete manner. 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.