Step |
Hyp |
Ref |
Expression |
1 |
|
idn1 |
⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ▶ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ) |
2 |
|
bicom |
⊢ ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) ) |
3 |
|
imbi2 |
⊢ ( ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) ) → ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ) ) |
4 |
3
|
biimpcd |
⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ) ) |
5 |
1 2 4
|
e10 |
⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ▶ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ) |
6 |
|
3impexp |
⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ) |
7 |
6
|
biimpi |
⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ) |
8 |
5 7
|
e1a |
⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ▶ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ) |
9 |
8
|
in1 |
⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ) |
10 |
|
idn1 |
⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ▶ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ) |
11 |
6
|
biimpri |
⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ) |
12 |
10 11
|
e1a |
⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ▶ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) ) |
13 |
3
|
biimprcd |
⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜏 ↔ 𝜃 ) ) → ( ( ( 𝜃 ↔ 𝜏 ) ↔ ( 𝜏 ↔ 𝜃 ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ) ) |
14 |
12 2 13
|
e10 |
⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ▶ ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ) |
15 |
14
|
in1 |
⊢ ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ) |
16 |
|
impbi |
⊢ ( ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) → ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ) → ( ( ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) → ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ) → ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ) ) ) |
17 |
9 15 16
|
e00 |
⊢ ( ( ( 𝜑 ∧ 𝜓 ∧ 𝜒 ) → ( 𝜃 ↔ 𝜏 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜒 → ( 𝜏 ↔ 𝜃 ) ) ) ) ) |