A wireless body area network (WBAN) has recently emerged as a new generation of wireless area networks. Mobile WBANs constitute an active field of research and development as it offers the potential of great improvement pervasive health monitoring and management services. However, the equipment used in WBAN is usually mobile with a constraint on energy. The energy efficiency must be taken into accounts as one of the objectives of the routing protocol designed for this type of network. Although the nodes are mobile causing links failures, most of studies ignore the link stability. In this paper, we propose a stable, reliable and energy efficient routing protocol for mobile Wireless Body Area Networks. It preserves the residual energy of nodes ensuring a reliable transfer of medical data. To achieve this goal, we use an objective model to select energy-efficient paths with stable links. A fuzzy logic system is used for link stability determination. Simulation results show improved performance of our proposed protocol in comparison to the selected existing protocol in terms of energy consumption, routing overhead, packet delivery ratio and end-to-end delay.