Consider the following premises. forall x_5: P_5(x_5) -> U_3(x_5). forall x_5: P_5(x_5). forall x_5: P_4(x_5) <-> U_3(x_5). Can we infer exists x_5: ~P_4(x_5) from them? If possible, name the inference rules being used at each step.
 From forall x_5: P_5(x_5) -> U_3(x_5), forall x_5: P_5(x_5) we can infer forall x_5: U_3(x_5) via universal modus ponens. From forall x_5: P_4(x_5) <-> U_3(x_5) we can infer forall x_5: U_3(x_5) -> P_4(x_5) via universal biconditional elimination. Finally, from forall x_5: U_3(x_5) -> P_4(x_5), forall x_5: U_3(x_5) we can infer forall x_5: P_4(x_5) via universal modus ponens, which contradicts exists x_5: ~P_4(x_5). Therefore, the answer is no.