Counting the number of bipartite matchings is #P-hard. Thus every #SAT problem can be reduced to counting the number of bipartite matchings. If a SAT problem is unsatisfiable however, it will have 0 solutions and thus the corresponding bipartite matching problem will have 0 solutions which is detectable in polynomial time (since a single bipartite matching can be found in polynomial time).
This would enable us to solve SAT in poly-time. What's going wrong with this reasoning?