Consider the following premises. For all x_1, if x_1 were a carpenter, then x_1 is driving a car. For all x_1, if x_1 drives a car, then x_1 is an artist. For all x_1, x_1 is a carpenter if and only if x_1 is an artist. For all x_1, if x_1 were an artist, then x_1 is a carpenter. Can we infer the following from them? For all x_1, x_1 is a carpenter if and only if x_1 is an artist.
That is one of the premises. Therefore, the answer is yes.