Step |
Hyp |
Ref |
Expression |
1 |
|
merlem12 |
⊢ ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) |
2 |
|
merlem12 |
⊢ ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) |
3 |
|
merlem5 |
⊢ ( ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) → ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) ) |
4 |
2 3
|
ax-mp |
⊢ ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) |
5 |
|
merlem6 |
⊢ ( ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) → ( ( ( ( ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) ) |
6 |
4 5
|
ax-mp |
⊢ ( ( ( ( ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) |
7 |
|
meredith |
⊢ ( ( ( ( ( ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) → ( ¬ ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → ( ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) ) |
8 |
6 7
|
ax-mp |
⊢ ( ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) |
9 |
1 8
|
ax-mp |
⊢ ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) |
10 |
|
merlem6 |
⊢ ( ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) ) ) |
11 |
9 10
|
ax-mp |
⊢ ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) ) |
12 |
|
merlem11 |
⊢ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) ) → ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) ) |
13 |
11 12
|
ax-mp |
⊢ ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) |
14 |
|
meredith |
⊢ ( ( ( ( ( 𝜓 → 𝜓 ) → ( ¬ 𝜑 → ¬ ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) ) ) → 𝜑 ) → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) ) ) |
15 |
13 14
|
ax-mp |
⊢ ( ( 𝜑 → 𝜓 ) → ( ( ( 𝜃 → ( ¬ ¬ 𝜒 → 𝜒 ) ) → ¬ ¬ 𝜑 ) → 𝜓 ) ) |