Static analysis plays a crucial role in program optimization, bug detection, and automated testing. Dyck-reachability provides a foundational formulation for static analysis, as Dyck grammars can model critical properties such as field and context sensitivity, thus offering broad applicability. This paper shows that static analysis problems modeled as Dyck-reachability on bidirected graphs can be encoded into the EUF SMT theory; consequently, all such problems admit efficient formulation and solution via EUF-based SMT solvers. By leveraging the optimized nature of modern SMT solvers, our method achieves efficiency comparable to state-of-the-art graph-based bidirected Dyck-reachability algorithms while eliminating the need for developing complex specialized graph reachability algorithms. Our approach opens new avenues for solving these classical static analysis problems, demonstrating the strong potential of SMT solvers in encoding static analysis solutions.