Methods exploiting problem symmetries have been very successful in several areas including constraint programming and SAT solving. We here recast a technique to enhance the performance of SMT-solvers by detecting symmetries in the input formulas and use them to prune the search space of the SMT algorithm. This technique is based on the concept of (syntactic) invariance by permutation of constants. An algorithm for solving SMT by taking advantage of such symmetries is presented. The implementation of this algorithm in the SMT-solver \veriT is used to illustrate the practical benefits of this approach. It results in a significant improvement of \veriT's performances on the SMT-LIB benchmarks that places it ahead of the winners of the last editions of the SMT-COMP contest in the QF\_UF category.