Corollary 2
Tag 04S8, permanent. Cite it as [Bourbaki, Tag 04S8].
Commutative Algebra, chapter VI, § 1. Valuation rings
For an integral domain to be integrally closed, it is necessary and sufficient that it be the intersection of a family of valuation rings of its field of fractions.
In the case of a Noetherian ring, Corollary 2 can be made more precise (Chapter VII, § 1, no. 3, Corollary to Theorem 1).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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