Consider the following premises. forall x_3: P_3(x_3) -> U_4(x_3). forall x_3: P_3(x_3) -> ~Q_3(x_3). forall x_3: U_4(x_3) -> P_3(x_3). forall x_3: Q_3(x_3). Can we infer exists x_3: U_4(x_3) from them? If possible, name the inference rules being used at each step.
 From forall x_3: P_3(x_3) -> ~Q_3(x_3), forall x_3: Q_3(x_3) we can infer forall x_3: ~P_3(x_3) via universal modus tollens. From forall x_3: U_4(x_3) -> P_3(x_3), forall x_3: P_3(x_3) -> U_4(x_3) we can infer forall x_3: U_4(x_3) <-> P_3(x_3) via universal biconditional introduction. Finally, from forall x_3: U_4(x_3) <-> P_3(x_3), forall x_3: ~P_3(x_3) we can infer forall x_3: ~U_4(x_3) via universal biconditional elimination, which contradicts exists x_3: U_4(x_3). Therefore, the answer is no.