Prove (P → Q) ↔ (¬Q → ¬P) using conditional elimination and negation introduction.

Question

I'm trying to prove that (P → Q) ↔ (¬Q → ¬P) using Fitch.

I know I have to prove two subproofs.

1) P → Q 2)¬Q → ¬P

Answer 1

Hint

For the first part :

1) P → Q --- premise

2) ¬Q --- assumed [a]

3) P --- assumed [b]

4) Q --- from 1) and 3) by →-elimination

5) ⊥ from 2) and 4)

6) ¬P --- from 3) and 5) by ¬-introduction, discharging [b]

7) ¬Q → ¬P --- from 2) and 6) by →-introduction, discharging [a].

The other part of the proof is similar.