Related Concepts
Various model theoretic ideas are related to quantifier elimination, and there are various equivalent conditions.
Every theory with quantifier elimination is model complete.
A first-order theory T has quantifier elimination if and only if for any two models B and C of T and for any common substructure A of B and C, B and C are elementarily equivalent in the language of T augmented with constants from A. In fact, it is sufficient here to show that any sentence with only existential quantifiers have the same truth value in B and C.
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