Consider the following premises. forall y: Q_2(y) -> U(y). exists y: Q_4(y) <-> P_2(y). forall y: ~U(y). forall y: Q_4(y) <-> Q_2(y). Can we infer exists y: ~P_2(y) from them? If possible, name the inference rules being used at each step.
 From forall y: Q_2(y) -> U(y), forall y: ~U(y) we can infer forall y: ~Q_2(y) via universal modus tollens. From forall y: Q_4(y) <-> Q_2(y), forall y: ~Q_2(y) we can infer forall y: ~Q_4(y) via universal biconditional elimination. Finally, from exists y: Q_4(y) <-> P_2(y), forall y: ~Q_4(y) we can infer exists y: ~P_2(y) via existential biconditional elimination. Therefore, the answer is yes.