We investigate the phenomenon of first-order transition [explosive synchronization (ES)] in an adaptively coupled phase-frustrated bilayer multiplex network. We consider Sakaguchi-Kuramoto dynamics over the top of multiplex networks and we establish that ES can emerge in all layers of a multiplex network even when one of the layers may not exhibit ES in the absence of the interlayer connections. We clearly identify the regions of the parameter space in which the multiplexity wins over the frustration parameter and network structure for the emergence of ES. Based on the mean-field analysis around the coherent state and a perturbative approach around the incoherent state we analytically derive the synchronization transition points (backward and forward) of all layers of the multiplex network as well as its monolayer counterpart satisfying a close agreement with the numerical results.