 Asymptotic equivalence of formulas

Two formulas with the same free variables are asymptotically equivalent when the probability of their universal equivalence tends to one over the chosen finite-structure distribution. If a quantifier-eliminating extension theory has all its axioms of asymptotic probability one, compactness makes every formula asymptotically equivalent to a quantifier-free one: each particular equivalence follows from finitely many axioms, whose conjunction still has limiting probability one.