Model checking is a useful technique to verify properties of dynamic systems but it has to cope with the state explosion problem. By simultaneous exploitation of symmetries of both the system and the property, the model checking can be performed on a reduced quotient structure [2,6,7]. In these techniques a property is specified within a temporal logic formula (CTL*) and the symmetries of the formula are obtained by a syntactical checking. We show here that these approaches fail to capture symmetries in the LTL path subformulas. Thus we propose a more accurate method based on local symmetries of the associated Büchi automaton. We define an appropriate quotient structure for the synchronized product of the Büchi automaton and the global state transition graph. We prove that model checking can be performed over this quotient structure leading to efficient algorithms.