Consider the following premises. For all x_1, x_1 will cook if and only if x_1 will drink water. There is at least one x_1 for which x_1 will drink water if and only if x_1 will cook. Can we infer the following from them? For all x_1, if is not true that x_1 will drink water if and only if x_1 will cook.
That contradicts the premises. Therefore the answer is no.