Definition 4
Tag 0C1K, permanent. Cite it as [Bourbaki, Tag 0C1K].
General Topology, chapter IX, § 1. Generation of a uniformity by a family of pseudometrics; uniformizable spaces
A topological space is said to be completely regular if it is uniformizable and Hausdorff.
Equivalently, in view of Theorem 2, a space is completely regular if it satisfies axioms (H) [cf. Chapter I, § 8, no. 1, Proposition 1] and (O_{IV}).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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Nothing.