**Proof.**
We will prove the lemma by induction on $d = \dim (\mathcal{O}_{X, x}) = \dim (\mathcal{O}_{Y, y})$.

An uninteresting case is when $d = 1$. In that case we are assuming that $f$ is unramified at $x$ and that $\mathcal{O}_{Y, y}$ is a discrete valuation ring (Algebra, Lemma 10.119.7). Then $\mathcal{O}_{X, x}$ is flat over $\mathcal{O}_{Y, y}$ (otherwise the map would not be quasi-finite at $x$) and we see that $f$ is flat at $x$. Since flat $+$ unramified is étale we conclude (some details omitted).

The case $d \geq 2$. We will use induction on $d$ to reduce to the case discussed in Lemma 58.21.3. To check $f$ is étale at $x$ we may work étale locally on the base and on the target (Descent, Lemmas 35.23.29 and 35.31.1). Thus we can apply More on Morphisms, Lemma 37.41.1 and assume that $f : X \to Y$ is finite and that $x$ is the unique point of $X$ lying over $y$. Here we use that étale extensions of local rings do not change dimension, normality, and regularity, see More on Algebra, Section 15.44 and Étale Morphisms, Section 41.19.

Next, we can base change by $\mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y})$ and assume that $Y$ is the spectrum of a regular local ring. It follows that $X = \mathop{\mathrm{Spec}}(\mathcal{O}_{X, x})$ as every point of $X$ necessarily specializes to $x$.

The ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is finite and necessarily injective (by equality of dimensions). We conclude we have going down for $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ by Algebra, Proposition 10.38.7 (and the fact that a regular ring is a normal ring by Algebra, Lemma 10.157.5). Pick $x' \in X$, $x' \not= x$ with image $y' = f(x')$. Then $\mathcal{O}_{X, x'}$ is normal as a localization of a normal domain. Similarly, $\mathcal{O}_{Y, y'}$ is regular (see Algebra, Lemma 10.110.6). We have $\dim (\mathcal{O}_{X, x'}) = \dim (\mathcal{O}_{Y, y'})$ by Algebra, Lemma 10.112.7 (we checked going down above). Of course these dimensions are strictly less than $d$ as $x' \not= x$ and by induction on $d$ we conclude that $f$ is étale at $x'$.

Thus we arrive at the following situation: We have a finite local homomorphism $A \to B$ of Noetherian local rings of dimension $d \geq 2$, with $A$ regular, $B$ normal, which induces a finite étale morphism $V \to U$ on punctured spectra. Our goal is to show that $A \to B$ is étale. This follows from Lemma 58.21.3 and the proof is complete.
$\square$

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