Consider the following premises. forall y: Q(y) -> U_4(y). forall y: U_4(y) -> Q(y). forall y: ~Q_2(y). exists y: Q(y) <-> Q_2(y). Can we infer exists y: ~U_4(y) from them? If possible, name the inference rules being used at each step.
 From forall y: Q(y) -> U_4(y), forall y: U_4(y) -> Q(y) we can infer forall y: Q(y) <-> U_4(y) via universal biconditional introduction. From exists y: Q(y) <-> Q_2(y), forall y: ~Q_2(y) we can infer exists y: ~Q(y) via existential biconditional elimination. Finally, from forall y: Q(y) <-> U_4(y), exists y: ~Q(y) we can infer exists y: ~U_4(y) via existential biconditional elimination. Therefore, the answer is yes.