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.
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 .
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 above. As with the rules for disjunction, these rules require that the variable quantified in one subformula does not appear free in the other subformula.
The rules for removing quantifiers from the antecedent are:
is equivalent to ,
is equivalent to .
The rules for removing quantifiers from the consequent are:
The rules for converting a formula to prenex form make heavy use of classical logic. In intuitionistic logic, it is not true that every formula is logically equivalent to a prenex formula. The negation connective is one obstacle, but not the only one. The implication operator is also treated differently in intuitionistic logic than classical logic; in intuitionistic logic, it is not definable using disjunction and negation.
The BHK interpretation illustrates why some formulas have no intuitionistically-equivalent prenex form. In this interpretation, a proof of
is a function which, given a concrete x and a proof of , produces a concrete y and a proof of ψ(y). In this case it is allowable for the value of y to be computed from the given value of x. A proof of
on the other hand, produces a single concrete value of y and a function that converts any proof of into a proof of ψ(y). If each x satisfying φ can be used to construct a y satisfying ψ but no such y can be constructed without knowledge of such an x then formula (1) will not be equivalent to formula (2).
The rules for converting a formula to prenex form that do fail in intuitionistic logic are:
(1) implies ,
(2) implies ,
(3) implies ,
(4) implies ,
(5) implies ,
(x does not appear as a free variable of in (1) and (3); x does not appear as a free variable of in (2) and (4)).