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