Definition 4
Tag 06KW, permanent. Cite it as [Bourbaki, Tag 06KW].
Algebra, chapter V, § 7. Separable algebraic extensions
— A field K is said to be separably closed if every separable algebraic extension of K is trivial.
An algebraically closed field is separably closed. Conversely, if a perfect field K is separably closed, it is algebraically closed, because every algebraic extension of K is separable (V, p. 3 , Prop. 2).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.