In this article, we derive the existence and uniqueness of continuous viscosity solution of Bellman-Isaacs system of equations, associated with a multiple mode switching game in finite horizon when the state process is constrained to live in a connected bounded closed domain. We rely on systems of penalized reflected generalized BSDEs with unilateral obstacles, that provide approximation of the system of PDEs. We obtain a comparison result which gives the uniqueness and the continuity of the viscosity solution. This allows us to use Perron's method to obtain the existence.