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God I feel like an idiot for questioning this, but is it really true not B implies not A in that situation? It seems like it depends on what you mean by "implies."

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.

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If A implies B (A -> B) then you cannot have A -> C Unless you meant it can imply both, but then still A implies B.

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

A dog can have 3 or less legs.

"not B" doesn't mean "something else which is not identically B", it means "the negation of B". Also, the statement "A->B" has absolutely no bearing on the statement "B->A". Hope this helps.

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