Sentential Logic Derivation Equivalence Question

Question

I am having trouble with the following question:

Construct a derivation to show the following two sentences are equivalent in SD.

1. ~(A≡B) 2. (~A≡B)

I am stuck at trying to introduce the material biconditional in order to find 2, because I have no clue how to derive A from B and vice versa with only sentence 1 as my premise.

Am I looking at this the wrong way? How would I go about doing this?

Answer 1

1. $\lnot (A \equiv B) \qquad\qquad$ Premise

2. $\qquad \lnot A \qquad\qquad \quad$ Assumption

3. $\qquad \qquad \lnot B \qquad \quad$ Assumption

4. $\qquad \qquad \qquad A \qquad$ Assumption

5. $\qquad \qquad \qquad \bot \qquad \bot Intro \: 2,4$

6. $\qquad \qquad \qquad B \qquad \bot Elim \: 5$

7. $\qquad \qquad \qquad B \qquad$ Assumption

8. $\qquad \qquad \qquad \bot \qquad \bot Intro \: 3,7$

9. $\qquad \qquad \qquad A \qquad \bot Elim \: 8$

10. $\qquad \qquad A \equiv B \qquad \equiv Intro \: 4-6,7-9$

11. $\qquad \qquad \bot \qquad \qquad \bot Intro \: 1,10$

12. $\qquad \lnot \lnot B \qquad\qquad \quad \lnot Intro \: 3-11$

13. $ \qquad B \qquad \qquad\qquad \lnot Elim \: 12$

14. $ \qquad B \qquad \qquad\qquad $ Assumption

15. $ \qquad \qquad A \qquad \qquad$ Assumption

16. $\qquad \qquad \qquad A \qquad$ Assumption

17. $\qquad \qquad \qquad B \qquad Reit \: 14$

18. $\qquad \qquad \qquad B \qquad$ Assumption

19. $\qquad \qquad \qquad A \qquad Reit \: 15$

20. $\qquad \qquad A \equiv B \qquad \equiv Intro \: 16-17,18-19$

21. $\qquad \qquad \bot \qquad \qquad \bot Intro \: 1,20$

22. $\qquad \lnot A \qquad\qquad \quad \lnot Intro \: 15-21$

23. $\lnot A \equiv B \qquad \qquad\quad\equiv Intro \: 2-13,14-22$