We present new constraint qualification conditions for nonlinear semidefinite programming that extend some of the constant rank-type conditions from nonlinear programming. As an application of these conditions, we provide a unified global convergence proof of a class of algorithms to stationary points without assuming neither uniqueness of the Lagrange multiplier nor boundedness of the Lagrange multipliers set. This class of algorithms includes, for instance, general forms of augmented Lagrangian, sequential quadratic programming, and interior point methods. In particular, we do not assume boundedness of the dual sequence generated by the algorithm. The weaker sequential condition we present is shown to be strictly weaker than Robinson’s condition while still implying metric subregularity.