Consider the following premises. forall x_10: T_4(x_10) <-> Q_10(x_10). forall x_10: ~Q_10(x_10). forall x_10: U_2(x_10) <-> T_4(x_10). exists x_10: W_4(x_10) -> U_2(x_10). forall x_10: U_2(x_10) -> W_4(x_10). Can we infer exists x_10: ~W_4(x_10) from them? If possible, name the inference rules being used at each step.
 From forall x_10: T_4(x_10) <-> Q_10(x_10), forall x_10: ~Q_10(x_10) we can infer forall x_10: ~T_4(x_10) via universal biconditional elimination. From forall x_10: U_2(x_10) -> W_4(x_10), exists x_10: W_4(x_10) -> U_2(x_10) we can infer exists x_10: U_2(x_10) <-> W_4(x_10) via existential biconditional introduction. From forall x_10: U_2(x_10) <-> T_4(x_10), forall x_10: ~T_4(x_10) we can infer forall x_10: ~U_2(x_10) via universal biconditional elimination. Finally, from exists x_10: U_2(x_10) <-> W_4(x_10), forall x_10: ~U_2(x_10) we can infer exists x_10: ~W_4(x_10) via existential biconditional elimination. Therefore, the answer is yes.