  In this paper, we provide proofs of two ${}_5\psi_5$ summation formulas of
Bailey using a ${}_5\phi_4$ identity of Carlitz. We show that in the limiting
case, the two ${}_5\psi_5$ identities give rise to two ${}_3\psi_3$ summation
formulas of Bailey. Finally, we prove the two ${}_3\psi_3$ identities using a
technique initially used by Ismail to prove Ramanujan's ${}_1\psi_1$ summation
formula and later by Ismail and Askey to prove Bailey's very-well-poised
${}_6\psi_6$ sum.
