Step |
Hyp |
Ref |
Expression |
1 |
|
luk-1 |
⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) |
2 |
|
luklem5 |
⊢ ( ¬ ( 𝜑 → 𝜓 ) → ( ¬ 𝜓 → ¬ ( 𝜑 → 𝜓 ) ) ) |
3 |
|
luklem2 |
⊢ ( ( ¬ 𝜓 → ¬ ( 𝜑 → 𝜓 ) ) → ( ( ( ¬ 𝜓 → 𝜓 ) → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) ) |
4 |
|
luklem4 |
⊢ ( ( ( ( ¬ 𝜓 → 𝜓 ) → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) |
5 |
3 4
|
luklem1 |
⊢ ( ( ¬ 𝜓 → ¬ ( 𝜑 → 𝜓 ) ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) |
6 |
2 5
|
luklem1 |
⊢ ( ¬ ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) |
7 |
|
luk-1 |
⊢ ( ( ¬ ( 𝜑 → 𝜓 ) → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) ) |
8 |
6 7
|
ax-mp |
⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) |
9 |
|
luk-1 |
⊢ ( ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) ) ) |
10 |
8 9
|
ax-mp |
⊢ ( ( ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) ) |
11 |
|
luklem4 |
⊢ ( ( ( ( ¬ ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) ) → ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) ) |
12 |
10 11
|
ax-mp |
⊢ ( ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) |
13 |
1 12
|
luklem1 |
⊢ ( ( 𝜑 → ( 𝜑 → 𝜓 ) ) → ( 𝜑 → 𝜓 ) ) |