Consider the following premises. forall x_15: P_11(x_15) or W(x_15). exists x_15: S(x_15) <-> W(x_15). forall x_15: ~P_15(x_15). forall x_15: P_11(x_15) -> ~Q_15(x_15). forall x_15: P_15(x_15) or Q_15(x_15). Can we infer exists x_15: S(x_15) from them? If possible, name the inference rules being used at each step.
 From forall x_15: P_15(x_15) or Q_15(x_15), forall x_15: ~P_15(x_15) we can infer forall x_15: Q_15(x_15) via universal disjunctive syllogism. From forall x_15: P_11(x_15) -> ~Q_15(x_15), forall x_15: Q_15(x_15) we can infer forall x_15: ~P_11(x_15) via universal modus tollens. From exists x_15: S(x_15) <-> W(x_15) we can infer exists x_15: W(x_15) -> S(x_15) via existential biconditional elimination. From forall x_15: P_11(x_15) or W(x_15), forall x_15: ~P_11(x_15) we can infer forall x_15: W(x_15) via universal disjunctive syllogism. Finally, from exists x_15: W(x_15) -> S(x_15), forall x_15: W(x_15) we can infer exists x_15: S(x_15) via existential modus ponens. Therefore, the answer is yes.