Consider the following premises. forall x_9: P(x_9) -> U(x_9). forall x_9: P(x_9) -> W_3(x_9). forall x_9: U(x_9) -> P(x_9). forall x_9: U(x_9) or W_1(x_9). forall x_9: W_1(x_9) -> P(x_9). Can we infer exists x_9: ~W_3(x_9) from them? If possible, name the inference rules being used at each step.
 From forall x_9: P(x_9) -> U(x_9), forall x_9: U(x_9) -> P(x_9) we can infer forall x_9: P(x_9) <-> U(x_9) via universal biconditional introduction. From forall x_9: P(x_9) <-> U(x_9) we can infer forall x_9: U(x_9) -> P(x_9) via universal biconditional elimination. From forall x_9: U(x_9) -> P(x_9), forall x_9: W_1(x_9) -> P(x_9), forall x_9: U(x_9) or W_1(x_9) we can infer forall x_9: P(x_9) via universal disjunction elimination. Finally, from forall x_9: P(x_9) -> W_3(x_9), forall x_9: P(x_9) we can infer forall x_9: W_3(x_9) via universal modus ponens, which contradicts exists x_9: ~W_3(x_9). Therefore, the answer is no.