Consider the following premises. forall y: P(y) -> U(y). exists y: ~U(y). forall y: U(y) <-> P(y). Can we infer forall y: P(y) from them?
 From forall y: U(y) <-> P(y) we can infer forall y: U(y) -> P(y). From forall y: U(y) -> P(y), forall y: P(y) -> U(y) we can infer forall y: U(y) <-> P(y). Finally, from forall y: U(y) <-> P(y), exists y: ~U(y) we can infer exists y: ~P(y), which contradicts forall y: P(y). Therefore, the answer is no.