Step |
Hyp |
Ref |
Expression |
1 |
|
tbw-ax4 |
⊢ ( ⊥ → 𝜒 ) |
2 |
|
tbw-ax1 |
⊢ ( ( 𝜓 → ⊥ ) → ( ( ⊥ → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) |
3 |
|
tbwlem1 |
⊢ ( ( ( 𝜓 → ⊥ ) → ( ( ⊥ → 𝜒 ) → ( 𝜓 → 𝜒 ) ) ) → ( ( ⊥ → 𝜒 ) → ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) ) ) |
4 |
2 3
|
ax-mp |
⊢ ( ( ⊥ → 𝜒 ) → ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) ) |
5 |
1 4
|
ax-mp |
⊢ ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) |
6 |
|
tbwlem1 |
⊢ ( ( ( 𝜓 → ⊥ ) → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( ( 𝜓 → ⊥ ) → 𝜒 ) ) ) |
7 |
5 6
|
ax-mp |
⊢ ( 𝜓 → ( ( 𝜓 → ⊥ ) → 𝜒 ) ) |
8 |
|
tbw-ax1 |
⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( ( ( 𝜓 → ⊥ ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) |
9 |
|
tbw-ax1 |
⊢ ( ( 𝜓 → ( ( 𝜓 → ⊥ ) → 𝜒 ) ) → ( ( ( ( 𝜓 → ⊥ ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) ) |
10 |
7 8 9
|
mpsyl |
⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) |
11 |
|
tbw-ax1 |
⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) ) |
12 |
10 11
|
tbwsyl |
⊢ ( ( 𝜑 → ( 𝜓 → ⊥ ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜃 ) → ( 𝜓 → 𝜃 ) ) ) |