In this paper, we consider recognizer P systems with antimatter and the influence of the matter/antimatter annihilation rules having weak priority over all the other rules or not. We first provide a uniform family of P systems with active membranes which solves the strongly NP-complete problem SAT, the Satisfiability Problem, without polarizations and without dissolution, yet with division for elementary membranes and with matter/antimatter annihilation rules having weak priority over all the other rules. Then we show that without this weak priority of the matter/antimatter annihilation rules over all the other rules we only obtain the complexity class P.