We unify search-based and compilation-based approaches to multi-agent path finding (MAPF) through satisfiability modulo theories (SMT). The task in MAPF is to navigate agents in an undirected graph to given goal vertices so that they do not collide. We rephrase Conflict-Based Search (CBS), one of the state-of-the-art algorithms for optimal MAPF solving, in the terms of SMT. This idea combines SAT-based solving known from MDD-SAT, a SAT-based optimal MAPF solver, at the low-level with conflict elimination of CBS at the high-level. Where the standard CBS branches the search after a conflict, we refine the propositional model with a disjunctive constraint. Our novel algorithm called SMT-CBS hence does not branch at the high-level but incrementally extends the propositional model. We experimentally compare SMT-CBS with CBS, ICBS, and MDD-SAT.