Consider the following premises. forall x: Q(x) <-> R_1(x). forall x: ~U(x). exists x: S_3(x) -> U(x). forall x: W(x) <-> S_3(x). forall x: R_1(x) <-> W(x). Can we infer exists x: ~Q(x) from them? If possible, name the inference rules being used at each step.
 From exists x: S_3(x) -> U(x), forall x: ~U(x) we can infer exists x: ~S_3(x) via existential modus tollens. From forall x: W(x) <-> S_3(x), exists x: ~S_3(x) we can infer exists x: ~W(x) via existential biconditional elimination. From forall x: R_1(x) <-> W(x), exists x: ~W(x) we can infer exists x: ~R_1(x) via existential biconditional elimination. Finally, from forall x: Q(x) <-> R_1(x), exists x: ~R_1(x) we can infer exists x: ~Q(x) via existential biconditional elimination. Therefore, the answer is yes.