Abstract Yang-Baxter deformations of superstring σ-models have recently inspired a supergravity solution generating technique. Using the open/closed string map and a Killing bi-vector as a deformation parameter, new solutions can be built, such that the (generalised) supergravity field equations were conjectured to always reduce to the classical Yang-Baxter equation (CYBE) [1]. In this work we provide a proof of this conjecture, using a systematic approach based on the so-called β-supergravity, which is a dynamical theory for the field β mn instead of the NSNS 2-form b mn .