Consider the following premises. exists x: ~W(x). forall x: W(x) -> Q(x). forall x: Q(x) or ~U_2(x). forall x: Q(x) -> W(x). Can we infer forall x: U_2(x) from them? If possible, name the inference rules being used at each step.
 From forall x: Q(x) -> W(x), forall x: W(x) -> Q(x) we can infer forall x: Q(x) <-> W(x) via universal biconditional introduction. From forall x: Q(x) <-> W(x), exists x: ~W(x) we can infer exists x: ~Q(x) via existential biconditional elimination. From forall x: Q(x) or ~U_2(x) we can infer forall x: U_2(x) -> Q(x) via universal rewriting. Finally, from forall x: U_2(x) -> Q(x), exists x: ~Q(x) we can infer exists x: ~U_2(x) via existential modus tollens, which contradicts forall x: U_2(x). Therefore, the answer is no.