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In this paper, we evaluate the performance of a vehicle-to-vehicle (V2V) system where full-duplex relay (FDR) harvests the energy from source and uses decode-and-forward (DF) protocol to forward data from source to destination. Unlike existing works about FDR systems, we consider the scenario that both relay and destination are moving vehicles, leading to the channel between relay and destination characterized by double (cascade) Rayleigh fading. We successfully obtain the closed-form mathematical expressions of the outage probability (OP) and throughput of the considered energy harvesting- (EH-) FDR-V2V system. Based on these expressions, the system performance is investigated through various scenarios. Numerical results indicate that the performance of the considered system is reduced compared with that of the system over Rayleigh fading channels. We also observe that there is an optimal EH time duration that minimizes the OP and maximizes the throughput. This value depends on the transmission power of source. Furthermore, the OP goes to outage floor faster due to the impact of the residual self-interference (RSI), especially when RSI is high. All analysis results are verified by Monte-Carlo simulations.