In this article we prove a new elliptic hypergeometric integral identity. It
previously appeared (as a conjecture) in articles by Rains, and Spiridonov and
Vartanov. Moreover it gives a different proof of an identity in another article
by Rains. We also give some basic hypergeometric and classical limits of this
identity. The classical limit gives identities (some known, some new) between
generalizations of the Selberg integral.