Constraint qualifications (CQs) are fundamental for understanding the geometry of feasible sets and for ensuring the validity of optimality conditions in nonlinear programming. A known idea is that constant-rank type CQs allow one to modify the description feasible set, by eliminating redundant constraints, so that the Mangasarian-Fromovitz CQ (MFCQ) holds. Traditionally, such modifications, called reductions here, have served primarily as auxiliary tools to connect existing CQs. In this work, we adopt a different viewpoint: we treat the very existence of such reductions as a CQ in itself. We study these “reduction-induced” CQs in a general framework, relating them not only to MFCQ, but also to arbitrary CQs. Moreover, we establish their connection with the local error bound (LEB) property. Building on this, we introduce a relaxed variant of the constant rank CQ known as constant rank of the subspace component (CRSC). This new CQ preserves the main geometric features of CRSC, guarantees LEB and the existence of reductions to MFCQ. Finally, we partially prove a recent conjecture stated in (SIAM J. Optim., 34(2):1799-1825, 2024) by showing that CRSC implies LEB in the manifold setting.