[Wikipedia states](https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(geometry)#Generalizations_and_related_results) that A. D. Alexandrov generalized *Cauchy's rigidity theorem for polyhedra* to higher dimensions. 


The relevant statement in the article is not linked to any source. The sources at the end of the Wikipedia page seem to be only about $3$-dimensional polyhedra as well, in particular Alexandrov's book "Convex polyhedra".



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Alexandrov's generalization of Cauchy's rigidity theorem
The following is Theorem 27.2 of Igor Pak's book [Lectures on Discrete and Polyhedral Geometry](http://www.math.ucla.edu/~pak/book.htm) (which in general is a very nice resource for these sorts of questions):



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> Let $P,Q\subset\mathbb{R}^d$ (or $P,Q \subset S^d\_+$), $d\geq3$ be two combinatorially equivalent (spherical) convex polyhedra whose corresponding facets are isometric. Then $P$ and $Q$ are isometric.
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(Here $S^d\_+$ is a d-dimensional hemisphere.)


