Consider the following premises. forall x: T_1(x) <-> W_2(x). exists x: T_1(x) -> W_4(x). forall x: W_4(x) -> T_1(x). forall x: ~T(x). forall x: W_2(x) -> T(x). Can we infer exists x: ~W_4(x) from them? If possible, name the inference rules being used at each step.
 From forall x: W_4(x) -> T_1(x), exists x: T_1(x) -> W_4(x) we can infer exists x: W_4(x) <-> T_1(x) via existential biconditional introduction. From forall x: W_2(x) -> T(x), forall x: ~T(x) we can infer forall x: ~W_2(x) via universal modus tollens. From forall x: T_1(x) <-> W_2(x), forall x: ~W_2(x) we can infer forall x: ~T_1(x) via universal biconditional elimination. Finally, from exists x: W_4(x) <-> T_1(x), forall x: ~T_1(x) we can infer exists x: ~W_4(x) via existential biconditional elimination. Therefore, the answer is yes.