Consider the following premises. forall x: U_1(x). exists x: U_1(x) -> P(x). forall x: U(x) or ~P(x). Can we infer forall x: ~U(x) from them? If possible, name the inference rules being used at each step.
 From forall x: U(x) or ~P(x) we can infer forall x: P(x) -> U(x) via universal rewriting. From exists x: U_1(x) -> P(x), forall x: P(x) -> U(x) we can infer exists x: U_1(x) -> U(x) via existential transitivity. Finally, from exists x: U_1(x) -> U(x), forall x: U_1(x) we can infer exists x: U(x) via existential modus ponens, which contradicts forall x: ~U(x). Therefore, the answer is no.