structural:reformatting, structural:summarization
Use of prenex form

Some proof calculi will only deal with a theory whose formulae are written in prenex normal form. The concept is essential for developing the arithmetical hierarchy and the analytical hierarchy.

Gödel 's proof of his completeness theorem for first-order logic presupposes that all formulae have been recast in prenex normal form.

Tarski's axioms for geometry is a logical system whose sentences can all be written in universal–existential form, a special case of the prenex normal form that has every universal quantifier preceding any existential quantifier, so that all sentences can be rewritten in the form ∀ u {\displaystyle \forall u} ∀ v {\displaystyle \forall v} … {\displaystyle \ldots } ∃ a {\displaystyle \exists a} ∃ b {\displaystyle \exists b} ϕ {\displaystyle \phi }, where ϕ {\displaystyle \phi } is a sentence that does not contain any quantifier. This fact allowed Tarski to prove that Euclidean geometry is decidable.