Consider the following premises. forall x_4: P_4(x_4) <-> R(x_4). forall x_4: R(x_4) or W(x_4). forall x_4: W(x_4) -> P_4(x_4). Can we infer exists x_4: ~P_4(x_4) from them? If possible, name the inference rules being used at each step.
 From forall x_4: P_4(x_4) <-> R(x_4) we can infer forall x_4: R(x_4) -> P_4(x_4) via universal biconditional elimination. Finally, from forall x_4: R(x_4) -> P_4(x_4), forall x_4: W(x_4) -> P_4(x_4), forall x_4: R(x_4) or W(x_4) we can infer forall x_4: P_4(x_4) via universal disjunction elimination, which contradicts exists x_4: ~P_4(x_4). Therefore, the answer is no.