Consider the following premises. For all z, if z were a carpenter, then z is playing a game. For all z, z will play a game if and only if z is a carpenter. There is at least one z for which if z were a carpenter, then z is playing a game. Can we infer the following from them? For all z, if is not true that z is a carpenter if and only if z will play a game.
That contradicts the premises. Therefore the answer is no.