Consider the following premises. forall y: U_1(y). forall y: R(y) <-> P(y). forall y: P(y) -> ~U_1(y). forall y: R(y) <-> S(y). Can we infer exists y: S(y) from them? If possible, name the inference rules being used at each step.
 From forall y: P(y) -> ~U_1(y), forall y: U_1(y) we can infer forall y: ~P(y) via universal modus tollens. From forall y: R(y) <-> P(y), forall y: ~P(y) we can infer forall y: ~R(y) via universal biconditional elimination. Finally, from forall y: R(y) <-> S(y), forall y: ~R(y) we can infer forall y: ~S(y) via universal biconditional elimination, which contradicts exists y: S(y). Therefore, the answer is no.