An improved prenex normal form1

Abstract

Let ℒ be the set of all formulas of a given first order predicate logic (with or without identity). For each positive integer n, let ℒn be the set of all formulas φ in ℒ logically equivalent to a formula of the form where Q is a (possibly empty) string of quantifiers, m is a positive integer, and each αij is either an atomic formula or the negation of an atomic formula.