In new wireless network scenarios, such as the future Internet of Things, it is necessary to transmit information and energy simultaneously, and the communication environment is complex and variable, with high maintenance costs for communication equipment. On the one hand, IRS can improve the performance of wireless communication and wireless power transfer by intelligent adjustment of the wireless channel. On the other hand, providing power supply to IRS by wireless power transfer can solve the problem of inconvenient power supply of IRS and reduce the maintenance costs. This paper investigated the throughput optimization of a dual wireless powered IRS-assisted wireless communication system, in which the multi-antenna base station(BS) first transmitted energy to two IRS, and then transmitted information to multiple single-antenna users with the help of the IRS. For the cases without and with channel blocking between the BS and the IRS, this paper proposed two-phase and three-phase transmission modes, respectively. It maximized the throughputs of these modes by optimizing the transmit beamforming of the BS, the reflect beamforming of the IRS, and the time allocation of different phases, subject to the constraints of the maximum transmit power of the BS, the minimum rate of the users, the duration of each phase, and the unit-module reflection coefficients of the IRS. Since the resulting optimization problems were convex and the optimization variables were highly coupled, it was difficult to find their optimal solution directly. This paper proposed two optimization algorithms based on the alternating optimization, successive convex approximation, and semi-definite relaxation techniques to obtain high-quality sub-optimal solutions. Simulation results show that the proposed algorithms achieve significantly better throughput performance compared to some benchmark algorithms and demonstrate the suitable application scenarios of the two-phase and three-phase transmission modes.