Consider the following premises. forall x_4: W(x_4) or ~R(x_4). forall x_4: ~R(x_4). exists x_4: R(x_4) -> W(x_4). forall x_4: W(x_4) -> R(x_4). Can we infer exists x_4: ~W(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: W(x_4) or ~R(x_4) we can infer forall x_4: R(x_4) -> W(x_4) via universal rewriting. From forall x_4: R(x_4) -> W(x_4), forall x_4: W(x_4) -> R(x_4) we can infer forall x_4: R(x_4) <-> W(x_4) via universal biconditional introduction. From forall x_4: R(x_4) <-> W(x_4) we can infer forall x_4: W(x_4) -> R(x_4) via universal biconditional elimination. From forall x_4: W(x_4) -> R(x_4), exists x_4: R(x_4) -> W(x_4) we can infer exists x_4: W(x_4) <-> R(x_4) via existential biconditional introduction. Finally, from exists x_4: W(x_4) <-> R(x_4), forall x_4: ~R(x_4) we can infer exists x_4: ~W(x_4) via existential biconditional elimination. Therefore, the answer is yes.