So, you're asked to produce a contradiction out of the following assumptions:

1.

2.

Forgive me for changing notation, but I hate the horseshoe for implication. I can never remember which way it's supposed to go!

I would go about it this way:

1. First, assume .

2. Then, show that .

3. Second, assume .

4. Then, show that .

5. The law of the excluded middle (which in natural deduction you can derive from scratch!) tells you that .

6. Therefore, you can conclude that .

Alternatively, you can go like this:

1. . Law of excluded middle.

2. by equivalence with step 1.

I'm just following the outline of the (valid) proof you've already provided. I think once you get started, you'll be able to continue.

So, how would you write this up so far in a two-column proof? And how would you continue?