Consider the following premises. forall x_2: U_2(x_2) -> Q(x_2). forall x_2: Q(x_2) -> U_2(x_2). exists x_2: Q(x_2) -> U_2(x_2). forall x_2: ~Q(x_2). Can we infer forall x_2: U_2(x_2) from them? If possible, name the inference rules being used at each step.
 From forall x_2: U_2(x_2) -> Q(x_2), forall x_2: Q(x_2) -> U_2(x_2) we can infer forall x_2: U_2(x_2) <-> Q(x_2) via universal biconditional introduction. From forall x_2: U_2(x_2) <-> Q(x_2) we can infer forall x_2: U_2(x_2) -> Q(x_2) via universal biconditional elimination. From exists x_2: Q(x_2) -> U_2(x_2), forall x_2: U_2(x_2) -> Q(x_2) we can infer exists x_2: Q(x_2) <-> U_2(x_2) via existential biconditional introduction. Finally, from exists x_2: Q(x_2) <-> U_2(x_2), forall x_2: ~Q(x_2) we can infer exists x_2: ~U_2(x_2) via existential biconditional elimination, which contradicts forall x_2: U_2(x_2). Therefore, the answer is no.