Consider the following premises. exists x: P(x) -> W(x). forall x: R_1(x) -> P(x). forall x: W(x) -> R_1(x). forall x: ~P(x). Can we infer exists x: ~R_1(x) from them? If possible, name the inference rules being used at each step.
 From exists x: P(x) -> W(x), forall x: W(x) -> R_1(x) we can infer exists x: P(x) -> R_1(x) via existential transitivity. From forall x: R_1(x) -> P(x), exists x: P(x) -> R_1(x) we can infer exists x: R_1(x) <-> P(x) via existential biconditional introduction. Finally, from exists x: R_1(x) <-> P(x), forall x: ~P(x) we can infer exists x: ~R_1(x) via existential biconditional elimination. Therefore, the answer is yes.