Step |
Hyp |
Ref |
Expression |
1 |
|
dfbi1 |
⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) |
2 |
|
pm2.21 |
⊢ ( ¬ ( 𝜓 → 𝜑 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) |
3 |
2
|
imim2i |
⊢ ( ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) → ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) ) |
4 |
|
id |
⊢ ( ¬ ( 𝜓 → 𝜑 ) → ¬ ( 𝜓 → 𝜑 ) ) |
5 |
|
falim |
⊢ ( ⊥ → ¬ ( 𝜓 → 𝜑 ) ) |
6 |
4 5
|
ja |
⊢ ( ( ( 𝜓 → 𝜑 ) → ⊥ ) → ¬ ( 𝜓 → 𝜑 ) ) |
7 |
6
|
imim2i |
⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) |
8 |
3 7
|
impbii |
⊢ ( ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ↔ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) ) |
9 |
8
|
notbii |
⊢ ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) ) |
10 |
|
pm2.21 |
⊢ ( ¬ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ⊥ ) ) |
11 |
|
ax-1 |
⊢ ( ¬ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ⊥ ) → ¬ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) ) ) |
12 |
|
falim |
⊢ ( ⊥ → ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ⊥ ) → ¬ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) ) ) |
13 |
11 12
|
ja |
⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ⊥ ) → ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ⊥ ) → ¬ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) ) ) |
14 |
13
|
pm2.43i |
⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ⊥ ) → ¬ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) ) |
15 |
10 14
|
impbii |
⊢ ( ¬ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) ↔ ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ⊥ ) ) |
16 |
1 9 15
|
3bitri |
⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ⊥ ) ) → ⊥ ) ) |