Definition 2
Tag 04S3, permanent. Cite it as [Bourbaki, Tag 04S3].
Commutative Algebra, chapter VI, § 1. Valuation rings
In the notation of Theorem 1, is called a valuation ring for the field if the equivalent conditions (a), (b), (c), (d), (e) hold. A ring is called a valuation ring if it is an integral domain and it is a valuation ring for its field of fractions.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.