Step |
Hyp |
Ref |
Expression |
1 |
|
merco2 |
⊢ ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) |
2 |
|
mercolem1 |
⊢ ( ( ( 𝜑 → ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) |
3 |
|
mercolem1 |
⊢ ( ( ( ( 𝜑 → ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) |
4 |
2 3
|
ax-mp |
⊢ ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) |
5 |
|
mercolem5 |
⊢ ( 𝜑 → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) |
6 |
|
mercolem4 |
⊢ ( ( 𝜑 → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) |
7 |
5 6
|
ax-mp |
⊢ ( ( ( 𝜑 → 𝜒 ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) |
8 |
4 7
|
ax-mp |
⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) |
9 |
1 8
|
ax-mp |
⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) |
10 |
|
mercolem1 |
⊢ ( ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) |
11 |
|
mercolem1 |
⊢ ( ( ( ( 𝜑 → ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) |
12 |
10 11
|
ax-mp |
⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) |
13 |
|
mercolem5 |
⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) |
14 |
|
mercolem4 |
⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) → ( ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) ) ) |
15 |
13 14
|
ax-mp |
⊢ ( ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) ) |
16 |
12 15
|
ax-mp |
⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) ) |
17 |
1 16
|
ax-mp |
⊢ ( ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) ) |
18 |
9 17
|
ax-mp |
⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) ) |
19 |
1 18
|
ax-mp |
⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) ) |
20 |
1 19
|
ax-mp |
⊢ ( ( ( ( 𝜑 → 𝜑 ) → ( ( ⊥ → 𝜑 ) → 𝜑 ) ) → ( ( 𝜑 → 𝜑 ) → ( 𝜑 → ( 𝜑 → 𝜑 ) ) ) ) → ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) |
21 |
1 20
|
ax-mp |
⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) |