Entanglement swapping entangles two particles that have never interacted[1], which implicitly assumes that each particle carries an independent local hidden variable, i.e., the presence of bilocality[2]. Previous experimental studies of bilocal hidden variable models did not fulfill the central requirement that the assumed two local hidden variable models must be mutually independent and hence their conclusions are flawed on the rejection of local realism[3-5]. By harnessing the laser phase randomization[6] rising from the spontaneous emission to the stimulated emission to ensure the independence between entangled photon-pairs created at separate sources and separating relevant events spacelike to satisfy the no-signaling condition, for the first time, we simultaneously close the loopholes of independent source, locality and measurement independence in an entanglement swapping experiment in a network. We measure a bilocal parameter of 1.181$pm$0.004 and the CHSH game value of 2.652$pm$0.059, indicating the rejection of bilocal hidden variable models by 45 standard deviations and local hidden variable models by 11 standard deviations. We hence rule out local realism and justify the presence of quantum nonlocality in our network experiment. Our experimental realization constitutes a fundamental block for a large quantum network. Furthermore, we anticipate that it may stimulate novel information processing applications[7,8].
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