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])
Q1x F[x] &or Q2x G[x] = Q1x Q2z F[x] &or G[z]
Q1x F[x] &and Q2x G[x] = Q1x Q2z F[x] &and G[z]