Consider the following premises. forall x_6: Q_6(x_6). forall x_6: P_6(x_6) <-> R(x_6). forall x_6: R(x_6) <-> U(x_6). forall x_6: P_6(x_6) -> ~Q_6(x_6). Can we infer exists x_6: U(x_6) from them? If possible, name the inference rules being used at each step.
 From forall x_6: P_6(x_6) -> ~Q_6(x_6), forall x_6: Q_6(x_6) we can infer forall x_6: ~P_6(x_6) via universal modus tollens. From forall x_6: P_6(x_6) <-> R(x_6), forall x_6: ~P_6(x_6) we can infer forall x_6: ~R(x_6) via universal biconditional elimination. Finally, from forall x_6: R(x_6) <-> U(x_6), forall x_6: ~R(x_6) we can infer forall x_6: ~U(x_6) via universal biconditional elimination, which contradicts exists x_6: U(x_6). Therefore, the answer is no.