Step |
Hyp |
Ref |
Expression |
1 |
|
dfbi1 |
⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ) |
2 |
|
imor |
⊢ ( ( 𝜑 → 𝜓 ) ↔ ( ¬ 𝜑 ∨ 𝜓 ) ) |
3 |
|
imor |
⊢ ( ( 𝜓 → 𝜑 ) ↔ ( ¬ 𝜓 ∨ 𝜑 ) ) |
4 |
3
|
notbii |
⊢ ( ¬ ( 𝜓 → 𝜑 ) ↔ ¬ ( ¬ 𝜓 ∨ 𝜑 ) ) |
5 |
2 4
|
imbi12i |
⊢ ( ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ↔ ( ( ¬ 𝜑 ∨ 𝜓 ) → ¬ ( ¬ 𝜓 ∨ 𝜑 ) ) ) |
6 |
5
|
notbii |
⊢ ( ¬ ( ( 𝜑 → 𝜓 ) → ¬ ( 𝜓 → 𝜑 ) ) ↔ ¬ ( ( ¬ 𝜑 ∨ 𝜓 ) → ¬ ( ¬ 𝜓 ∨ 𝜑 ) ) ) |
7 |
|
pm4.62 |
⊢ ( ( ( ¬ 𝜑 ∨ 𝜓 ) → ¬ ( ¬ 𝜓 ∨ 𝜑 ) ) ↔ ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( ¬ 𝜓 ∨ 𝜑 ) ) ) |
8 |
7
|
notbii |
⊢ ( ¬ ( ( ¬ 𝜑 ∨ 𝜓 ) → ¬ ( ¬ 𝜓 ∨ 𝜑 ) ) ↔ ¬ ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( ¬ 𝜓 ∨ 𝜑 ) ) ) |
9 |
1 6 8
|
3bitri |
⊢ ( ( 𝜑 ↔ 𝜓 ) ↔ ¬ ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ¬ ( ¬ 𝜓 ∨ 𝜑 ) ) ) |