 Theorem 5.15 (Feferman 1975; Jäger 1983; Jäger &amp; Pohlers 1982; Rathjen 1993b) \((\Delta^1_2{\Hy}\bCA)+\BI\), KPi, \(\bT_0\) and CZF+REA are proof-theoretically equivalent. In particular, these theories prove the same theorems in the negative arithmetic fragment.