Corollary 2
Tag 06KD, permanent. Cite it as [Bourbaki, Tag 06KD].
Algebra, chapter V, § 7. Separable algebraic extensions
— If an element x of E is algebraic and separable over K, it is algebraic and separable over every extension K' of K contained in E. Let f be the minimal polynomial of x over K. Then x is a simple root of f by Prop. 5, and since f belongs to K'[X], the element x of E is separable over K' by Cor. 1.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.