Consider the following premises. forall x_7: P(x_7) or ~U(x_7). exists x_7: P(x_7). forall x_7: P(x_7) -> U(x_7). Can we infer forall x_7: ~U(x_7) from them? If possible, name the inference rules being used at each step.
 From forall x_7: P(x_7) or ~U(x_7) we can infer forall x_7: U(x_7) -> P(x_7) via universal rewriting. From forall x_7: U(x_7) -> P(x_7), forall x_7: P(x_7) -> U(x_7) we can infer forall x_7: U(x_7) <-> P(x_7) via universal biconditional introduction. From forall x_7: U(x_7) <-> P(x_7) we can infer forall x_7: P(x_7) -> U(x_7) via universal biconditional elimination. Finally, from forall x_7: P(x_7) -> U(x_7), exists x_7: P(x_7) we can infer exists x_7: U(x_7) via existential modus ponens, which contradicts forall x_7: ~U(x_7). Therefore, the answer is no.