Consider the following premises. forall x_1: T_4(x_1) -> U(x_1). exists x_1: ~U(x_1). forall x_1: T_4(x_1) or ~U(x_1). Can we infer forall x_1: T_4(x_1) from them? If possible, name the inference rules being used at each step.
 From forall x_1: T_4(x_1) or ~U(x_1) we can infer forall x_1: U(x_1) -> T_4(x_1) via universal rewriting. From forall x_1: U(x_1) -> T_4(x_1), forall x_1: T_4(x_1) -> U(x_1) we can infer forall x_1: U(x_1) <-> T_4(x_1) via universal biconditional introduction. Finally, from forall x_1: U(x_1) <-> T_4(x_1), exists x_1: ~U(x_1) we can infer exists x_1: ~T_4(x_1) via existential biconditional elimination, which contradicts forall x_1: T_4(x_1). Therefore, the answer is no.