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