Let F be a 2CNF formula. A can be found.
Every proof is a path through the directed graph induced by the axioms.
Thus finding a strongly connected component containg a variable and its negation means that the theory is contradictory and has no model. Compress all equivalent variables by computing strongly connected components. This results in a theory in which no two variables are equvalent. Stop if you got a contradiction. For each variable and its negation, compute the directed transitive closure.