Step |
Hyp |
Ref |
Expression |
1 |
|
nannan |
⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜒 ) ) ↔ ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) ) |
2 |
1
|
biimpi |
⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜒 ) ) → ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) ) |
3 |
|
simpr |
⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜒 ) |
4 |
3
|
imim2i |
⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) → ( 𝜑 → 𝜒 ) ) |
5 |
|
simpl |
⊢ ( ( 𝜓 ∧ 𝜒 ) → 𝜓 ) |
6 |
5
|
imim2i |
⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) → ( 𝜑 → 𝜓 ) ) |
7 |
|
pm2.27 |
⊢ ( 𝜑 → ( ( 𝜑 → 𝜓 ) → 𝜓 ) ) |
8 |
7
|
anim2d |
⊢ ( 𝜑 → ( ( 𝜃 ∧ ( 𝜑 → 𝜓 ) ) → ( 𝜃 ∧ 𝜓 ) ) ) |
9 |
8
|
expdimp |
⊢ ( ( 𝜑 ∧ 𝜃 ) → ( ( 𝜑 → 𝜓 ) → ( 𝜃 ∧ 𝜓 ) ) ) |
10 |
6 9
|
syl5com |
⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) → ( ( 𝜑 ∧ 𝜃 ) → ( 𝜃 ∧ 𝜓 ) ) ) |
11 |
|
ancr |
⊢ ( ( 𝜑 → 𝜒 ) → ( 𝜑 → ( 𝜒 ∧ 𝜑 ) ) ) |
12 |
11
|
anim1i |
⊢ ( ( ( 𝜑 → 𝜒 ) ∧ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜃 ∧ 𝜓 ) ) ) → ( ( 𝜑 → ( 𝜒 ∧ 𝜑 ) ) ∧ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜃 ∧ 𝜓 ) ) ) ) |
13 |
4 10 12
|
syl2anc |
⊢ ( ( 𝜑 → ( 𝜓 ∧ 𝜒 ) ) → ( ( 𝜑 → ( 𝜒 ∧ 𝜑 ) ) ∧ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜃 ∧ 𝜓 ) ) ) ) |
14 |
|
con3 |
⊢ ( ( ( 𝜑 ∧ 𝜃 ) → ( 𝜃 ∧ 𝜓 ) ) → ( ¬ ( 𝜃 ∧ 𝜓 ) → ¬ ( 𝜑 ∧ 𝜃 ) ) ) |
15 |
|
df-nan |
⊢ ( ( 𝜃 ⊼ 𝜓 ) ↔ ¬ ( 𝜃 ∧ 𝜓 ) ) |
16 |
|
df-nan |
⊢ ( ( 𝜑 ⊼ 𝜃 ) ↔ ¬ ( 𝜑 ∧ 𝜃 ) ) |
17 |
14 15 16
|
3imtr4g |
⊢ ( ( ( 𝜑 ∧ 𝜃 ) → ( 𝜃 ∧ 𝜓 ) ) → ( ( 𝜃 ⊼ 𝜓 ) → ( 𝜑 ⊼ 𝜃 ) ) ) |
18 |
17
|
anim2i |
⊢ ( ( ( 𝜑 → ( 𝜒 ∧ 𝜑 ) ) ∧ ( ( 𝜑 ∧ 𝜃 ) → ( 𝜃 ∧ 𝜓 ) ) ) → ( ( 𝜑 → ( 𝜒 ∧ 𝜑 ) ) ∧ ( ( 𝜃 ⊼ 𝜓 ) → ( 𝜑 ⊼ 𝜃 ) ) ) ) |
19 |
|
nannan |
⊢ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜑 ) ) ↔ ( 𝜑 → ( 𝜒 ∧ 𝜑 ) ) ) |
20 |
19
|
biimpri |
⊢ ( ( 𝜑 → ( 𝜒 ∧ 𝜑 ) ) → ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜑 ) ) ) |
21 |
|
nanim |
⊢ ( ( ( 𝜃 ⊼ 𝜓 ) → ( 𝜑 ⊼ 𝜃 ) ) ↔ ( ( 𝜃 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) |
22 |
21
|
biimpi |
⊢ ( ( ( 𝜃 ⊼ 𝜓 ) → ( 𝜑 ⊼ 𝜃 ) ) → ( ( 𝜃 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) |
23 |
20 22
|
anim12i |
⊢ ( ( ( 𝜑 → ( 𝜒 ∧ 𝜑 ) ) ∧ ( ( 𝜃 ⊼ 𝜓 ) → ( 𝜑 ⊼ 𝜃 ) ) ) → ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜑 ) ) ∧ ( ( 𝜃 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) ) |
24 |
2 13 18 23
|
4syl |
⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜒 ) ) → ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜑 ) ) ∧ ( ( 𝜃 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) ) |
25 |
|
nannan |
⊢ ( ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜒 ) ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜑 ) ) ⊼ ( ( 𝜃 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) ) ↔ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜒 ) ) → ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜑 ) ) ∧ ( ( 𝜃 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) ) ) |
26 |
24 25
|
mpbir |
⊢ ( ( 𝜑 ⊼ ( 𝜓 ⊼ 𝜒 ) ) ⊼ ( ( 𝜑 ⊼ ( 𝜒 ⊼ 𝜑 ) ) ⊼ ( ( 𝜃 ⊼ 𝜓 ) ⊼ ( ( 𝜑 ⊼ 𝜃 ) ⊼ ( 𝜑 ⊼ 𝜃 ) ) ) ) ) |