Quantifier Elimination

A procedure or property of a first-order theory or formal language by which every formula that may contain existential or universal quantifiers is transformed into an equivalent formula that contains no quantifiers, relative to the semantics of the theory.

A theory T (or a structure) has quantifier elimination when every first-order formula is equivalent modulo T to a quantifier-free formula; equivalently, truth of any formula is determined by quantifier-free information in models of T.

A property of a theory stating that every formula is equivalent in the theory to a quantifier-free formula; equivalently, definable sets can be described without existential or universal quantifiers in the given language.