Consider the following premises. forall x_14: ~P_14(x_14). exists x_14: P_14(x_14) -> W_1(x_14). forall x_14: P_14(x_14) <-> W_1(x_14). forall x_14: R_1(x_14) -> W_1(x_14). Can we infer exists x_14: ~R_1(x_14) from them? If possible, name the inference rules being used at each step.
 From forall x_14: P_14(x_14) <-> W_1(x_14) we can infer forall x_14: W_1(x_14) -> P_14(x_14) via universal biconditional elimination. From exists x_14: P_14(x_14) -> W_1(x_14), forall x_14: W_1(x_14) -> P_14(x_14) we can infer exists x_14: P_14(x_14) <-> W_1(x_14) via existential biconditional introduction. From exists x_14: P_14(x_14) <-> W_1(x_14), forall x_14: ~P_14(x_14) we can infer exists x_14: ~W_1(x_14) via existential biconditional elimination. Finally, from forall x_14: R_1(x_14) -> W_1(x_14), exists x_14: ~W_1(x_14) we can infer exists x_14: ~R_1(x_14) via existential modus tollens. Therefore, the answer is yes.