By Michael Norrish and Rene Vestergaard, from TPHOLs 2007, available from Michael Norrish’s website:
Placing our result in a web of related mechanised results, we give a direct proof that the de Bruijn λ-calculus (a la Huet, Nipkow and Shankar) is isomorphic to an α-quotiented λ-calculus. In order to establish the link, we introduce an “index-carrying” abstraction mechanism over de Bruijn terms, and consider it alongside a simplified substitution mechanism. Relating the new notions to those of the α-quotiented and the proper
de Bruijn formalisms draws on techniques from the theory of nominal sets.
Slides from a related presentation are available here.