Consider the following premises. forall y: P_4(y) <-> P_1(y). exists y: P_4(y) -> U(y). forall y: U(y) -> P_1(y). Can we infer exists y: P_4(y) -> P_1(y) from them? If possible, name the inference rules being used at each step.
 From forall y: P_4(y) <-> P_1(y) we can infer forall y: P_1(y) -> P_4(y) via universal biconditional elimination. Finally, from exists y: P_4(y) -> U(y), forall y: U(y) -> P_1(y) we can infer exists y: P_4(y) -> P_1(y) via existential transitivity. Therefore, the answer is yes.