I need help with part (b) of the following question:

Attempt:

To solve this I'm trying to follow a similar worked example here. And the definition of the formal system $\displaystyle K_L$ is here.

So, we know that $\displaystyle ((p \to q) \to (\neg q \to \neg p))$ is a tautology since:

$\displaystyle (p \to q) \iff (\neg p \vee q)$

$\displaystyle \iff (q \vee \neg p)$

$\displaystyle \iff (\neg \neg q \vee \neg p)$

$\displaystyle \iff (\neg q \to \neg p)$

Now, I guess the first step is to show that $\displaystyle \{ \forall x (A \to B), \neg(\forall x \neg B) \} \vdash_{K_L} \neg \forall x \neg A$.

1. $\displaystyle \forall x (A \to B)$ .....(Hyp)

2. $\displaystyle (\forall x (A \to B)) \to (\forall x (\neg B \to \neg A))$ .....(instance of tautology $\displaystyle ((p \to q) \to (\neg q \to \neg p))$)

3. $\displaystyle (\forall x (\neg B \to \neg A))$ .....(1,2, Modus Ponen)

4. ...

I'm stuck at this point... How should I continue? Any help is appreciated.