Conjunctive grammars (Okhotin, 2001) are an extension of the standard context-free grammars with a conjunction operation, which maintains most of their practical properties, including many parsing algorithms. This paper introduces a further extension to the model, which is equipped with quantifiers for referring to the left context, in which the substring being defined does occur. For example, a rule A → a & ◁B defines a string a, as long as it is preceded by any string defined by B. The paper gives two equivalent definitions of the model-by logical deduction and by language equations-and establishes its basic properties, including a transformation to a normal form, a cubic-time parsing algorithm, and another recognition algorithm working in linear space.