Corollary

Tag 0CZV, permanent. Cite it as [Bourbaki, Tag 0CZV].

Commutative Algebra, chapter IX, Appendix 0.

Suppose that A is noetherian. a) One has dim(A) = dim(B). b) Suppose that A is regular, and let $(x_1, ..., x_n)$ be a system of coordinates of A. Then B is regular and the sequence $(x_1 1_B, ..., x_n 1_B)$ is a system of coordinates of B. This follows from Prop. 1 of VIII, § 5, No. 1.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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