The conversion to DNF is fine. The original clauses map directly to pairs of new clauses, so fixing an assignment to the original variables V you can always sat

The conversion to DNF is fine. The original clauses map directly to pairs of new clauses, so fixing an assignment to the original variables V you can always satisfy the same number of the DNF clauses. You can do worse by choosing the wrong values for X, sure, but there's always a straightforward correspondence.