Tautology, contingent or contradiction?

We can construct the following truth table for each side of this biconditional proposition:

We can see that the last two columns are identical.

Therefore, the proposition in the problem is a $\boxed{\text{tautology}}$

By De Morgan's law,

$\neg(A \wedge \neg B) \leftrightarrow (\neg A \vee \neg \neg B) \leftrightarrow (\neg A \vee B)$

and by conditional disjunction,

$(\neg A \vee B) \leftrightarrow (A\to B)$

which is precisely what we want. Therefore, the propositional statement is a tautology .

An easy to look at it is using set notation:

$\neg(A \rightarrow B)$ using set's $A$ and $B \implies$ $\exists x \in A \wedge x \not \in B \implies$ $x \in A \cap \neg B$ $\implies \neg(A \rightarrow B) \leftrightarrow (A \wedge \neg B) \implies$ $(A \rightarrow B) \leftrightarrow \neg (A \wedge \neg B )$