I think it's a trick question, in the way he phrases it. If A implies B, then not B implies not A. But he doesn't say that. He says, "If A implies B, does not A imply not B," for which the answer is no.
E.g., you could have A -> B, and A -> C, and B != C. Then C is not B, but implies A just as much as B might (in the very least it doesn't imply not A per se, as A might be true). It seems like there's some implicit assumptions going on.
A dog (A) has 4 legs(B). Something that does not have 4 legs (not B) is not a dog (not A). A cow though (not A) could still have 4 legs