Prenex normal form

A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in propositional logic (e.g. disjunctive normal form or conjunctive normal form), it provides a canonical normal form useful in automated theorem proving. Every formula in classical logic is logically equivalent to a formula in prenex normal form. For example, if , , and are quantifier-free formulas with the free variables shown then is in prenex normal form with matrix , while is logically equivalent but not in prenex normal form. Every first-order formula is logically equivalent (in classical logic) to some formula in prenex normal form. There are several conversion rules that can be recursively applied to convert a formula to prenex normal form. The rules depend on which logical connectives appear in the formula. The rules for conjunction and disjunction say that is equivalent to under (mild) additional condition , or, equivalently, (meaning that at least one individual exists), is equivalent to ; and is equivalent to , is equivalent to under additional condition . The equivalences are valid when does not appear as a free variable of ; if does appear free in , one can rename the bound in and obtain the equivalent . For example, in the language of rings, is equivalent to , but is not equivalent to because the formula on the left is true in any ring when the free variable x is equal to 0, while the formula on the right has no free variables and is false in any nontrivial ring. So will be first rewritten as and then put in prenex normal form . The rules for negation say that is equivalent to and is equivalent to . There are four rules for implication: two that remove quantifiers from the antecedent and two that remove quantifiers from the consequent. These rules can be derived by rewriting the implication as and applying the rules for disjunction and negation above.