Natural Deduction proof question

Question

Questions on Natural deduction proof:

1.(A→A) → (B→B)

2.(B→C) → (A→A) / conclusion: (B→B)

I was able to solve it using indirect proof but I want to try to prove it using the rules of inference and rules of replacement:

1.(A→A) → (B→B)

2.(B→C) → (A→A)

3.~ (A→A) → ~(B→C) 2 Contra

4.~ (B→B) → ~(A→A) 1 Contra

5.~ (B→B) → ~(B→C) 3, 4 HS

6. ~~(B→B) v ~(B→C) 5 Imp

then I am stuck. please help with the question. Thank you!

Answer 1

You're well on your way:

From 6):

$6. \ \neg \neg (B \rightarrow B) \lor \neg (B \rightarrow C)$

$7. \ (B \rightarrow B) \lor \neg (B \rightarrow C) \quad 6, DN$

$8. \ (B \rightarrow B) \lor \neg (\neg B \lor C) \quad 7, Impl$

$9. \ (B \rightarrow B) \lor (\neg \neg B \land \neg C) \quad 8, DeM$

$10. \ (B \rightarrow B) \lor (B \land \neg C) \quad 9, DN$

$11. \ ((B \rightarrow B) \lor B) \land ((B \rightarrow B) \lor \neg C) \quad 10, Dist$

$12. \ (B \rightarrow B) \lor B \quad 11, Simp$

$13. \ (\neg B \lor B) \lor B \quad 12, Impl$

$14. \ \neg B \lor (B \lor B) \quad 13, Assoc$

$15. \ \neg B \lor B \quad 14, Taut$

$16. \ B \rightarrow B \quad 15, Impl$