Satisfiabililty sufficient condition?
taking the contrapositive, another stronger way of stating your question is "a SAT formula f is satisfiable iff either the sub-formulas f1, f2 from the assignment of a variable x or not x are satisfiable". but this is exactly the idea behind the DPLL algorithm (originating as the simpler DP algorithm) provably guaranteed to find a SAT solution. your basic idea also shows the similarity of the method with Gaussian reduction; ie this is not usually remarked but DPLL algorithm appears to be like/ an analog of "boolean Gaussian reduction".