Consider the following premises. forall x_12: P_12(x_12) <-> W(x_12). exists x_12: ~P_12(x_12). forall x_12: W(x_12) <-> R_4(x_12). Can we infer forall x_12: R_4(x_12) from them? If possible, name the inference rules being used at each step.
 From forall x_12: P_12(x_12) <-> W(x_12) we can infer forall x_12: W(x_12) -> P_12(x_12) via universal biconditional elimination. From forall x_12: W(x_12) -> P_12(x_12), exists x_12: ~P_12(x_12) we can infer exists x_12: ~W(x_12) via existential modus tollens. Finally, from forall x_12: W(x_12) <-> R_4(x_12), exists x_12: ~W(x_12) we can infer exists x_12: ~R_4(x_12) via existential biconditional elimination, which contradicts forall x_12: R_4(x_12). Therefore, the answer is no.