Consider the following premises. For all x, x will play a game or x will not read a book. There is at least one x for which if x were a lawyer, then x is reading a book. For all x, if x plays a game, then x is a lawyer. Can we infer the following from them? For all x, if is not true that x will play a game if and only if x will read a book.
 From the fact that for all x, if x plays a game, then x is a lawyer, and that there is at least one x for which if x were a lawyer, then x is reading a book we can infer that there is at least one x for which if x plays a game, then x is reading a book. From the fact that for all x, x will play a game or x will not read a book we can infer that for all x, if x reads a book, then x is playing a game. Finally, from the fact that there is at least one x for which if x plays a game, then x is reading a book, and that for all x, if x reads a book, then x is playing a game we can infer that there is at least one x for which x will play a game if and only if x will read a book, which contradicts that For all x, if is not true that x will play a game if and only if x will read a book. Therefore, the answer is no.