We show the q-superlinear convergence of a reduced BFGS method for equality constrained problems, using eventually only one constraint linearization per iteration. The local method is globalized either with a standard arc-search or when an update criterion is satisfied, with a piecewise line-search. The aim of the latter technique is to realize generalized Wolfe conditions, which allow the algorithm to maintain naturally the positive definiteness of the generated matrices. We show that if the sequence of iterates converges, the convergence is q-superlinear. No assumption is made on the speed of convergence of the sequence of iterates or on the boundedness of the sequence of generated matrices. The main difficulty is to show that the ideal step-size is accepted after finitely many steps.