We study the properties of a polymer brush exposed to a mixture of two solvents where one component, called the co-nonsolvent (CNS), has a stronger preference with respect to the polymer. The concept of preferential adsorption of CNS onto the polymer, as recently proposed by Mukherhi, Kremer, and Marques [Nat. Commun. 2014, 5, 4882], is combined with the mean-field Alexander-de Gennes approach for the polymer brush. The key assumption is that CNS can form bridges between two monomers which is associated with a further gain in free energy, thus leading to an effective monomer-monomer attraction. The adsorption equilibrium of CNS at a given value of the monomer concentration results in a concentration-dependent chi-function for the polymer brush which describes the effective interactions between the monomers in the mixed solvent. This in turn can lead to a discontinuous collapse transition and to a corresponding reentry transition at higher CNS concentrations. The problem can be analytically treated for a minimal model where the increase of self-volume of the monomers due to adsorption of CNS is neglected. In this case the collapse and the reswelling transition have the same signature. For low brush densities in the noncollapsed state we give an analytic approximation for the spinodal of the collapse. This also allows to define two scaling variables instead of the three control parameters which are the grafting density, the CNS-monomer selectivity, and the volume faction of CNS. The proposed effective free energy contribution resulting from the CNS adsorption equilibrium can be transferred to other systems such as to gels or dendrimers.