A predicate calculus formula is in prenex normal form if all the quantifiers are at the left.

A formula can be transformed to prenex normal form using the following
identities. The form *Qx* is used for either *&forall x* or *&exist x*.

*Qx F[x] &or G = Qx (F[x] &or G)*, x not used in G.

*Qx F[x] &and G = Qx (F[x] &and G)*, x not used in G.

*¬ &forall x F[x] = &exist x ¬ F[x]*

*¬ &exist x F[x] = &forall x ¬ F[x]*

*&forall x F[x] &and &forall x G[x] = &forall x (F[x] &and G[x])*

*&exist x F[x] &or &exist x G[x] = &exist x (F[x] &or G[x])*

*Q _{1}x F[x] &or Q_{2}x G[x] = Q_{1}x Q_{2}z F[x] &or G[z]*