Proposition 5
Tag 06KB, permanent. Cite it as [Bourbaki, Tag 06KB].
Algebra, chapter V, § 7. Separable algebraic extensions
— Let E be an extension of K, x an element of E algebraic over K and f the minimal polynomial of x over K. Then the following conditions are equivalent: a) x is separable over K; b) the polynomial f is separable; c) x is a simple root of f. The equivalence of a) and b) follows from Prop. 3, that of b) and c) from Prop. 3 and 4 (cf. V, p. 37 and 38).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.