0 |__                        [assumption, want P iff not not P]
1 |   |__ P                  [assumption, want not not P]
2 |   |   |__ not P          [assumption, for reductio]
3 |   |   |   contradiction  [contradiction introduction: 1, 2]
4 |   |   not not P          [negation introduction: 2]
  |
5 |   |__ not not P          [assumption, want P]
6 |   |   P                  [negation elimination: 5]
  |
7 |   P iff not not P        [biconditional introduction: 1 - 4, 5 - 6]

ergo go back to your big math books.