Consider the following premises. forall x_10: R(x_10) <-> Q_10(x_10). forall x_10: ~Q_10(x_10). forall x_10: R(x_10). exists x_10: W(x_10). Can we infer exists x_10: W(x_10) and S(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: R(x_10) <-> Q_10(x_10), forall x_10: ~Q_10(x_10) we can infer forall x_10: ~R(x_10) via universal biconditional elimination. From forall x_10: R(x_10) we can infer forall x_10: R(x_10) or S(x_10) via universal addition. From forall x_10: R(x_10) or S(x_10), forall x_10: ~R(x_10) we can infer forall x_10: S(x_10) via universal disjunctive syllogism. Finally, from exists x_10: W(x_10), forall x_10: S(x_10) we can infer exists x_10: W(x_10) and S(x_10) via existential conjunction. Therefore, the answer is yes. Notice, however, that the premises were contradictory, so we can infer anything from them.