We study a generalization of the classical notion of implication, called weak implication. It extends unquantified predicate logic with a single level of existential quantification. We present a sound and complete set of deduction rules for weak implications. The notion of weak implication was introduced for the sake of a formal specification of a symbolic constraint solving system. Other practical applications of can be found in the realm of relational database theory: query normalization and integrity constraints in the context of views.