Step |
Hyp |
Ref |
Expression |
1 |
|
re1ax2lem |
⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( 𝜓 → ( 𝜑 → 𝜒 ) ) ) |
2 |
|
tb-ax1 |
⊢ ( ( 𝜑 → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) ) |
3 |
|
tb-ax3 |
⊢ ( ( ( ( 𝜑 → 𝜒 ) → 𝜒 ) → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) |
4 |
2 3
|
tbsyl |
⊢ ( ( 𝜑 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) |
5 |
|
tb-ax1 |
⊢ ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → ( 𝜑 → 𝜒 ) ) ) ) |
6 |
|
re1ax2lem |
⊢ ( ( ( 𝜑 → 𝜓 ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → ( 𝜑 → 𝜒 ) ) ) ) → ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( 𝜑 → 𝜒 ) ) ) ) ) |
7 |
5 6
|
ax-mp |
⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( 𝜑 → 𝜒 ) ) ) ) |
8 |
|
tb-ax1 |
⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) ) |
9 |
|
re1ax2lem |
⊢ ( ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( 𝜑 → 𝜒 ) ) ) → ( ( ( 𝜑 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) ) → ( ( ( 𝜑 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) ) ) |
10 |
8 9
|
ax-mp |
⊢ ( ( ( 𝜑 → ( 𝜑 → 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) → ( ( ( 𝜑 → 𝜓 ) → ( 𝜑 → ( 𝜑 → 𝜒 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) ) |
11 |
4 7 10
|
mpsyl |
⊢ ( ( 𝜓 → ( 𝜑 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) |
12 |
1 11
|
tbsyl |
⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜑 → 𝜒 ) ) ) |