Consider the following premises. forall z: R_1(z) -> W_1(z). exists z: W_1(z) -> Q_4(z). forall z: R_1(z) or ~Q_4(z). Can we infer exists z: R_1(z) <-> Q_4(z) from them? If possible, name the inference rules being used at each step.
 From forall z: R_1(z) -> W_1(z), exists z: W_1(z) -> Q_4(z) we can infer exists z: R_1(z) -> Q_4(z) via existential transitivity. From forall z: R_1(z) or ~Q_4(z) we can infer forall z: Q_4(z) -> R_1(z) via universal rewriting. Finally, from exists z: R_1(z) -> Q_4(z), forall z: Q_4(z) -> R_1(z) we can infer exists z: R_1(z) <-> Q_4(z) via existential biconditional introduction. Therefore, the answer is yes.