In this paper we present a model-theoretic criterion for quantifier elimination being a variant of Shoenfield's theorem (see [1], Chap. V, [paragraph]5). Our short proof is based directly on Godel's completeness and compactness theorems as well as on the concept of diagrams, and does not involve model-completeness or Robinson's test as does for instance the proof of certain related criteria given in [2], Chap. VIII, [paragraph]4. As a consequence, we immediately obtain the theorems of Chevalley and Tarski-Seidenberg from algebraic and semialalgebraic geometry.