Deductive Logic : If A -> B it can be deduced neg(A) -> neg(B)

Question

I am having a hard time proving the following $$ (a \to b) \vdash (\lnot a \to \lnot b) $$

I followed the book advice and first proved that $ (a \to b) \vdash (\lnot \lnot a \to \lnot \lnot b) $ using the Deduction Theorem but I am stuck afterwards. I have been trying to use every Axioms/Modus/Rules I know but always end up with unsatisfactory result.

A hint would be appreciated,

Answer 1

If it is raining, then there are clouds out. ($a\implies b$)

If it is not raining, then there are no clouds out ($\neg a \implies \neg b$)

The first is true and the second isn't. There is something close to what you want, called contrapositive.