In this paper, Cline's formula for the well-known generalized inverses such as Drazin inverse, pseudo Drazin inverse and generalized Drazin inverse is extended to the case when $\big\{\!\begin{smallmatrix}acd=dbd\\ dba=aca\end{smallmatrix}$. Also, applications are given to some interesting Banach space operator properties like algebraic, meromorphic, polaroidness and B-Fredholmness.