Consider the following premises. exists z_1: W(z_1) -> Q(z_1). forall z_1: Q(z_1) -> W(z_1). forall z_1: W(z_1) -> Q(z_1). Can we infer exists z_1: Q(z_1) <-> W(z_1) from them?
 From exists z_1: W(z_1) -> Q(z_1), forall z_1: Q(z_1) -> W(z_1) we can infer exists z_1: W(z_1) <-> Q(z_1). From exists z_1: W(z_1) <-> Q(z_1) we can infer exists z_1: Q(z_1) -> W(z_1). Finally, from exists z_1: Q(z_1) -> W(z_1), forall z_1: W(z_1) -> Q(z_1) we can infer exists z_1: Q(z_1) <-> W(z_1). Therefore, the answer is yes.