Does $\lnot(A→B) ⊢ A→\lnot B$ in classical logic?

Question

Given $\lnot(A→B)$. Then $A˄\lnot B$. Assume $\lnot(A→\lnot B)$. Then $A˄\lnot\lnot B$. By DN, $A˄B$. Also $A˄\lnot B (\text{reit})$. Contradiction, therefore, $\lnot\lnot(A→\lnot B)$ by indirect proof, which by DN is $(A→\lnot B)$. So $\lnot (A→B) ⊢ A→\lnot B$.

Answer 1

In propositional logic the following statements are equivalent:

• $p\vdash q$
• $\vdash p\to q$

So if you are still in doubt then you can just check whether statement:$$\neg(A\to B)\to(A\to\neg B)$$is a tautology by means of a truth table (as suggested in the comments on your question).