Consider the following premises. For all x_11, x_11 will write a letter or x_11 will work. For all x_11, x_11 will sleep if and only if x_11 will work. For all x_11, x_11 will sleep or x_11 will not go running. There is at least one x_11 for which x_11 will not write a letter or x_11 will go running. Can we infer the following from them? For all x_11, x_11 will sleep if and only if x_11 will work.
That is one of the premises. Therefore, the answer is yes.