Consider the following premises. forall x_4: ~T(x_4). forall x_4: T(x_4) -> Q_4(x_4). forall x_4: R_4(x_4) -> T(x_4). exists x_4: Q_4(x_4) -> R_4(x_4). Can we infer forall x_4: R_4(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: T(x_4) -> Q_4(x_4), exists x_4: Q_4(x_4) -> R_4(x_4) we can infer exists x_4: T(x_4) -> R_4(x_4) via existential transitivity. From exists x_4: T(x_4) -> R_4(x_4), forall x_4: R_4(x_4) -> T(x_4) we can infer exists x_4: T(x_4) <-> R_4(x_4) via existential biconditional introduction. Finally, from exists x_4: T(x_4) <-> R_4(x_4), forall x_4: ~T(x_4) we can infer exists x_4: ~R_4(x_4) via existential biconditional elimination, which contradicts forall x_4: R_4(x_4). Therefore, the answer is no.