Consider the following premises. exists z: W_2(z) -> W_4(z). forall z: W_4(z) -> W_2(z). forall z: W_4(z) -> W_2(z). Can we infer forall z: ~(W_2(z) <-> W_4(z)) from them?
 From exists z: W_2(z) -> W_4(z), forall z: W_4(z) -> W_2(z) we can infer exists z: W_2(z) <-> W_4(z). From exists z: W_2(z) <-> W_4(z) we can infer exists z: W_2(z) -> W_4(z). Finally, from exists z: W_2(z) -> W_4(z), forall z: W_4(z) -> W_2(z) we can infer exists z: W_2(z) <-> W_4(z), which contradicts forall z: ~(W_2(z) <-> W_4(z)). Therefore, the answer is no.