Clause-elimination procedures that simplify formulas in conjunctive normal form play an important role in modern SAT solving. Before or during the actual solving process, such procedures identify and remove clauses that are irrelevant to the solving result. These simplifications usually rely on so-called redundancy properties that characterize cases in which the removal of a clause does not affect the satisfiability status of a formula. One particularly successful redundancy property is that of blocked clauses, because it generalizes several other redundancy properties. To find out whether a clause is blocked---and therefore redundant---one only needs to consider its resolution environment, i.e., the clauses with which it can be resolved. For this reason, we say that the redundancy property of blocked clauses is local. In this paper, we show that there exist local redundancy properties that are even more general than blocked clauses. We present a semantic notion of blocking and prove that it constitutes the most general local redundancy property. We furthermore introduce the syntax-based notions of set-blocking and super-blocking, and show that the latter coincides with our semantic blocking notion. In addition, we show how semantic blocking can be alternatively characterized via Davis and Putnam's rule for eliminating atomic formulas. Finally, we perform a detailed complexity analysis and relate our novel redundancy properties to prominent redundancy properties from the literature.