Consider the following premises. forall x: U_2(x) or ~P(x). forall x: U(x). forall x: P(x). Can we infer exists x: ~(U_2(x) and U(x)) from them? If possible, name the inference rules being used at each step.
 From forall x: U_2(x) or ~P(x) we can infer forall x: P(x) -> U_2(x) via universal rewriting. From forall x: P(x) -> U_2(x), forall x: P(x) we can infer forall x: U_2(x) via universal modus ponens. Finally, from forall x: U_2(x), forall x: U(x) we can infer forall x: U_2(x) and U(x) via universal conjunction, which contradicts exists x: ~(U_2(x) and U(x)). Therefore, the answer is no.