Consider the following premises. forall x: U(x) -> Q_2(x). forall x: Q_2(x) or W(x). forall x: W(x) -> T(x). forall x: U(x) -> T(x). forall x: Q_2(x) -> U(x). Can we infer exists x: ~T(x) from them? If possible, name the inference rules being used at each step.
 From forall x: U(x) -> Q_2(x), forall x: Q_2(x) -> U(x) we can infer forall x: U(x) <-> Q_2(x) via universal biconditional introduction. From forall x: U(x) <-> Q_2(x) we can infer forall x: Q_2(x) -> U(x) via universal biconditional elimination. From forall x: Q_2(x) -> U(x), forall x: U(x) -> T(x) we can infer forall x: Q_2(x) -> T(x) via universal transitivity. Finally, from forall x: Q_2(x) -> T(x), forall x: W(x) -> T(x), forall x: Q_2(x) or W(x) we can infer forall x: T(x) via universal disjunction elimination, which contradicts exists x: ~T(x). Therefore, the answer is no.