Consider the following premises. forall x_3: ~U(x_3). forall x_3: S(x_3) -> U(x_3). forall x_3: P(x_3) <-> S(x_3). Can we infer exists x_3: P(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: S(x_3) -> U(x_3), forall x_3: ~U(x_3) we can infer forall x_3: ~S(x_3) via universal modus tollens. From forall x_3: P(x_3) <-> S(x_3) we can infer forall x_3: P(x_3) -> S(x_3) via universal biconditional elimination. Finally, from forall x_3: P(x_3) -> S(x_3), forall x_3: ~S(x_3) we can infer forall x_3: ~P(x_3) via universal modus tollens, which contradicts exists x_3: P(x_3). Therefore, the answer is no.