Consider the following premises. forall x: P(x) or ~U(x). forall x: U(x) <-> P(x). exists x: ~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) <-> P(x) we can infer forall x: P(x) -> U(x) via universal biconditional elimination. From forall x: P(x) or ~U(x) we can infer forall x: U(x) -> P(x) via universal rewriting. From forall x: U(x) -> P(x), forall x: P(x) -> U(x) we can infer forall x: U(x) <-> P(x) via universal biconditional introduction. Finally, from forall x: U(x) <-> P(x), exists x: ~P(x) we can infer exists x: ~U(x) via existential biconditional elimination, which contradicts forall x: U(x). Therefore, the answer is no.