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