Consider the following premises. forall y: P_4(y) -> U(y). exists y: T(y) <-> Q(y). forall y: U(y) -> Q(y). forall y: ~U(y). forall y: Q(y) -> P_4(y). Can we infer exists y: ~T(y) from them? If possible, name the inference rules being used at each step.
 From forall y: Q(y) -> P_4(y), forall y: P_4(y) -> U(y) we can infer forall y: Q(y) -> U(y) via universal transitivity. From forall y: Q(y) -> U(y), forall y: U(y) -> Q(y) we can infer forall y: Q(y) <-> U(y) via universal biconditional introduction. From forall y: Q(y) <-> U(y), forall y: ~U(y) we can infer forall y: ~Q(y) via universal biconditional elimination. Finally, from exists y: T(y) <-> Q(y), forall y: ~Q(y) we can infer exists y: ~T(y) via existential biconditional elimination. Therefore, the answer is yes.