Step |
Hyp |
Ref |
Expression |
1 |
|
orc |
⊢ ( ¬ 𝜑 → ( ¬ 𝜑 ∨ 𝜓 ) ) |
2 |
|
olc |
⊢ ( 𝜓 → ( ¬ 𝜑 ∨ 𝜓 ) ) |
3 |
1 2
|
ja |
⊢ ( ( 𝜑 → 𝜓 ) → ( ¬ 𝜑 ∨ 𝜓 ) ) |
4 |
3
|
imim1i |
⊢ ( ( ( ¬ 𝜑 ∨ 𝜓 ) → ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) → ( ( 𝜑 → 𝜓 ) → ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) ) |
5 |
|
pm2.24 |
⊢ ( 𝜃 → ( ¬ 𝜃 → 𝜑 ) ) |
6 |
|
idd |
⊢ ( 𝜃 → ( 𝜑 → 𝜑 ) ) |
7 |
5 6
|
jaod |
⊢ ( 𝜃 → ( ( ¬ 𝜃 ∨ 𝜑 ) → 𝜑 ) ) |
8 |
7
|
com12 |
⊢ ( ( ¬ 𝜃 ∨ 𝜑 ) → ( 𝜃 → 𝜑 ) ) |
9 |
|
pm1.5 |
⊢ ( ( ¬ ( 𝜑 → 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) → ( 𝜒 ∨ ( ¬ ( 𝜑 → 𝜓 ) ∨ ( 𝜃 ∨ 𝜏 ) ) ) ) |
10 |
|
pm2.3 |
⊢ ( ( ¬ ( 𝜑 → 𝜓 ) ∨ ( 𝜃 ∨ 𝜏 ) ) → ( ¬ ( 𝜑 → 𝜓 ) ∨ ( 𝜏 ∨ 𝜃 ) ) ) |
11 |
|
pm1.5 |
⊢ ( ( ¬ ( 𝜑 → 𝜓 ) ∨ ( 𝜏 ∨ 𝜃 ) ) → ( 𝜏 ∨ ( ¬ ( 𝜑 → 𝜓 ) ∨ 𝜃 ) ) ) |
12 |
|
pm2.21 |
⊢ ( ¬ 𝜑 → ( 𝜑 → 𝜓 ) ) |
13 |
|
jcn |
⊢ ( 𝜃 → ( ¬ 𝜑 → ¬ ( 𝜃 → 𝜑 ) ) ) |
14 |
12 13
|
imim12i |
⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜃 ) → ( ¬ 𝜑 → ( ¬ 𝜑 → ¬ ( 𝜃 → 𝜑 ) ) ) ) |
15 |
14
|
pm2.43d |
⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜃 ) → ( ¬ 𝜑 → ¬ ( 𝜃 → 𝜑 ) ) ) |
16 |
15
|
con4d |
⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜃 ) → ( ( 𝜃 → 𝜑 ) → 𝜑 ) ) |
17 |
|
imor |
⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜃 ) ↔ ( ¬ ( 𝜑 → 𝜓 ) ∨ 𝜃 ) ) |
18 |
|
imor |
⊢ ( ( ( 𝜃 → 𝜑 ) → 𝜑 ) ↔ ( ¬ ( 𝜃 → 𝜑 ) ∨ 𝜑 ) ) |
19 |
16 17 18
|
3imtr3i |
⊢ ( ( ¬ ( 𝜑 → 𝜓 ) ∨ 𝜃 ) → ( ¬ ( 𝜃 → 𝜑 ) ∨ 𝜑 ) ) |
20 |
19
|
orim2i |
⊢ ( ( 𝜏 ∨ ( ¬ ( 𝜑 → 𝜓 ) ∨ 𝜃 ) ) → ( 𝜏 ∨ ( ¬ ( 𝜃 → 𝜑 ) ∨ 𝜑 ) ) ) |
21 |
|
pm1.5 |
⊢ ( ( 𝜏 ∨ ( ¬ ( 𝜃 → 𝜑 ) ∨ 𝜑 ) ) → ( ¬ ( 𝜃 → 𝜑 ) ∨ ( 𝜏 ∨ 𝜑 ) ) ) |
22 |
10 11 20 21
|
4syl |
⊢ ( ( ¬ ( 𝜑 → 𝜓 ) ∨ ( 𝜃 ∨ 𝜏 ) ) → ( ¬ ( 𝜃 → 𝜑 ) ∨ ( 𝜏 ∨ 𝜑 ) ) ) |
23 |
22
|
orim2i |
⊢ ( ( 𝜒 ∨ ( ¬ ( 𝜑 → 𝜓 ) ∨ ( 𝜃 ∨ 𝜏 ) ) ) → ( 𝜒 ∨ ( ¬ ( 𝜃 → 𝜑 ) ∨ ( 𝜏 ∨ 𝜑 ) ) ) ) |
24 |
|
pm1.5 |
⊢ ( ( 𝜒 ∨ ( ¬ ( 𝜃 → 𝜑 ) ∨ ( 𝜏 ∨ 𝜑 ) ) ) → ( ¬ ( 𝜃 → 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ) |
25 |
9 23 24
|
3syl |
⊢ ( ( ¬ ( 𝜑 → 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) → ( ¬ ( 𝜃 → 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ) |
26 |
|
imor |
⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) ↔ ( ¬ ( 𝜑 → 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) ) |
27 |
|
imor |
⊢ ( ( ( 𝜃 → 𝜑 ) → ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ↔ ( ¬ ( 𝜃 → 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ) |
28 |
25 26 27
|
3imtr4i |
⊢ ( ( ( 𝜑 → 𝜓 ) → ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) → ( ( 𝜃 → 𝜑 ) → ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ) |
29 |
4 8 28
|
syl2im |
⊢ ( ( ( ¬ 𝜑 ∨ 𝜓 ) → ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) → ( ( ¬ 𝜃 ∨ 𝜑 ) → ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ) |
30 |
|
imor |
⊢ ( ( ( ¬ 𝜑 ∨ 𝜓 ) → ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) ↔ ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) ) |
31 |
|
imor |
⊢ ( ( ( ¬ 𝜃 ∨ 𝜑 ) → ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ↔ ( ¬ ( ¬ 𝜃 ∨ 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ) |
32 |
29 30 31
|
3imtr3i |
⊢ ( ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) → ( ¬ ( ¬ 𝜃 ∨ 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ) |
33 |
32
|
imori |
⊢ ( ¬ ( ¬ ( ¬ 𝜑 ∨ 𝜓 ) ∨ ( 𝜒 ∨ ( 𝜃 ∨ 𝜏 ) ) ) ∨ ( ¬ ( ¬ 𝜃 ∨ 𝜑 ) ∨ ( 𝜒 ∨ ( 𝜏 ∨ 𝜑 ) ) ) ) |