We report the first implementation of the von Neumann instantaneous measurements of nonlocal variables, which becomes possible due to technological achievements in creating hyperentangled photons. Tests of reliability and of the nondemolition property of the measurements have been performed with high precision, showing the suitability of the scheme as a basic ingredient of numerous quantum information protocols. The method allows us to demonstrate for the first time with strong measurements a special feature of pre- and postselected quantum systems: the failure of the product rule. It has been verified experimentally that for a particular pre- and postselected pair of particles, a single measurement on particle A yields with certainty σ_{x}^{A}=-1, a single measurement on particle B yields with certainty σ_{y}^{B}=-1, and a single nonlocal measurement on particles A and B yields with certainty σ_{x}^{A}σ_{y}^{B}=-1.