Consider the following premises. forall z: P_3(z) -> U(z). forall z: U(z) -> P_3(z). Can we infer forall z: U(z) <-> P_3(z) from them? If possible, name the inference rules being used at each step.
We can infer forall z: U(z) <-> P_3(z) via the universal biconditional introduction rule. Therefore, the answer is yes.