Deductive Reasoning proof logic

Question

I have a question for a deductive reasoning proof i made for the following question: (i needed to prove that these three premises can come to the conclusion of : ~R <--> ~T Im new to this so im not completely sure that this is correct.

Premises 1-3

1. R--> (S-->T)
2. S
3. ~T

End of Premises --------

4. Sub proof Assumption S
5. Sub proof T
6. (end of sub proof) S-->T . -->Intro, 4-5
7. R . --> Elim 1,6
8. T -->Elim 2,6
9. Sub proof Assumption R
10. Sub proof T
11. Sub proof ~T
12. Sub proof ⊥
13. (end of sub proof) ~R . ~Intro, 9-11
14. Sub proof ~R
15. Sub proof ~T
16. New Sub proof ~T
17. New sub proof ~R
18. ~R <--> ~T <--> Intro 13-14, 15-16

Answer 1

We have to derive $\lnot R \to \lnot T$ and vice-versa; then conclude with $\leftrightarrow$-intro.

The first part is straightforward, From 3rd premise : $\lnot T$, using $\to$-intro we get immediately:

4) $\lnot R \to \lnot T$.

For the second part :

5) $R$ --- temporary assumed for a sub-proof

6) $T$ --- from 5) and 2nd premise from 1st one

7) $\bot$ --- contradicition of 6) with 3rd premise

8) $\lnot R$ --- from 5) and 8) by $\lnot$-intro, discharging temporary assumption.

9) $\lnot T \to \lnot R$ --- from 8) by $\to$-intro.

10) $\lnot R \leftrightarrow \lnot T$ --- from 4) and 9) by $\leftrightarrow$-intro.