Consider the following premises. exists x: ~R_1(x). forall x: W_2(x) <-> R_1(x). forall x: R_1(x) -> W_2(x). Can we infer exists x: ~W_2(x) from them? If possible, name the inference rules being used at each step.
 From forall x: W_2(x) <-> R_1(x) we can infer forall x: W_2(x) -> R_1(x) via universal biconditional elimination. From forall x: R_1(x) -> W_2(x), forall x: W_2(x) -> R_1(x) we can infer forall x: R_1(x) <-> W_2(x) via universal biconditional introduction. Finally, from forall x: R_1(x) <-> W_2(x), exists x: ~R_1(x) we can infer exists x: ~W_2(x) via existential biconditional elimination. Therefore, the answer is yes.