The current paper is devoted to automated techniques in the correspondence theory. The theory we deal with concerns the problem of finding classical first-order axioms corresponding to propositional modal schemas. Given a modal schema and a semantics base  method of translating propositional modal formulae into classical first-order ones, we try to derive automatically classica first-order formulae characterizing precisely the class of frames validating the schema. The technique we consider can, in many cases, be easily applied even without computer support. Although we mainly concentrate on Kripke semantics, the technique we apply is much more general, as it is based on elimination of second-order quantifiers from formulae. We show many examples of application of the method. These can also serve as new, automated proofs of considered correspondences.