Step | Hyp | Ref
| Expression |
1 | | equequ1 2033 |
. . . . . 6
⊢ (𝑎 = 𝑡 → (𝑎 = 𝑏 ↔ 𝑡 = 𝑏)) |
2 | | neeq1 3003 |
. . . . . 6
⊢ (𝑎 = 𝑡 → (𝑎 ≠ 𝑐 ↔ 𝑡 ≠ 𝑐)) |
3 | 1, 2 | 3anbi12d 1439 |
. . . . 5
⊢ (𝑎 = 𝑡 → ((𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ (𝑡 = 𝑏 ∧ 𝑡 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
4 | 3 | 2exbidv 1932 |
. . . 4
⊢ (𝑎 = 𝑡 → (∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑏∃𝑐(𝑡 = 𝑏 ∧ 𝑡 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
5 | 4 | cbvexvw 2045 |
. . 3
⊢
(∃𝑎∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑡∃𝑏∃𝑐(𝑡 = 𝑏 ∧ 𝑡 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
6 | 5 | a1i 11 |
. 2
⊢ (𝑎 = 𝑡 → (∃𝑎∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑡∃𝑏∃𝑐(𝑡 = 𝑏 ∧ 𝑡 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
7 | | equequ2 2034 |
. . . . . . 7
⊢ (𝑏 = 𝑎 → (𝑡 = 𝑏 ↔ 𝑡 = 𝑎)) |
8 | | neeq1 3003 |
. . . . . . 7
⊢ (𝑏 = 𝑎 → (𝑏 ≠ 𝑐 ↔ 𝑎 ≠ 𝑐)) |
9 | 7, 8 | 3anbi13d 1440 |
. . . . . 6
⊢ (𝑏 = 𝑎 → ((𝑡 = 𝑏 ∧ 𝑡 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ (𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐))) |
10 | 9 | exbidv 1929 |
. . . . 5
⊢ (𝑏 = 𝑎 → (∃𝑐(𝑡 = 𝑏 ∧ 𝑡 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑐(𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐))) |
11 | 10 | cbvexvw 2045 |
. . . 4
⊢
(∃𝑏∃𝑐(𝑡 = 𝑏 ∧ 𝑡 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑎∃𝑐(𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐)) |
12 | 11 | exbii 1855 |
. . 3
⊢
(∃𝑡∃𝑏∃𝑐(𝑡 = 𝑏 ∧ 𝑡 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑡∃𝑎∃𝑐(𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐)) |
13 | 12 | a1i 11 |
. 2
⊢ (𝑏 = 𝑎 → (∃𝑡∃𝑏∃𝑐(𝑡 = 𝑏 ∧ 𝑡 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ ∃𝑡∃𝑎∃𝑐(𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐))) |
14 | | equequ1 2033 |
. . . . . . 7
⊢ (𝑡 = 𝑏 → (𝑡 = 𝑎 ↔ 𝑏 = 𝑎)) |
15 | | neeq1 3003 |
. . . . . . 7
⊢ (𝑡 = 𝑏 → (𝑡 ≠ 𝑐 ↔ 𝑏 ≠ 𝑐)) |
16 | 14, 15 | 3anbi12d 1439 |
. . . . . 6
⊢ (𝑡 = 𝑏 → ((𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ (𝑏 = 𝑎 ∧ 𝑏 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐))) |
17 | 16 | 2exbidv 1932 |
. . . . 5
⊢ (𝑡 = 𝑏 → (∃𝑎∃𝑐(𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ ∃𝑎∃𝑐(𝑏 = 𝑎 ∧ 𝑏 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐))) |
18 | 17 | cbvexvw 2045 |
. . . 4
⊢
(∃𝑡∃𝑎∃𝑐(𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ ∃𝑏∃𝑎∃𝑐(𝑏 = 𝑎 ∧ 𝑏 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐)) |
19 | | excom 2166 |
. . . . 5
⊢
(∃𝑏∃𝑎∃𝑐(𝑏 = 𝑎 ∧ 𝑏 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ ∃𝑎∃𝑏∃𝑐(𝑏 = 𝑎 ∧ 𝑏 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐)) |
20 | | 3ancomb 1101 |
. . . . . . 7
⊢ ((𝑏 = 𝑎 ∧ 𝑏 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ (𝑏 = 𝑎 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
21 | | equcom 2026 |
. . . . . . . 8
⊢ (𝑏 = 𝑎 ↔ 𝑎 = 𝑏) |
22 | 21 | 3anbi1i 1159 |
. . . . . . 7
⊢ ((𝑏 = 𝑎 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) ↔ (𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
23 | 20, 22 | bitri 278 |
. . . . . 6
⊢ ((𝑏 = 𝑎 ∧ 𝑏 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ (𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
24 | 23 | 3exbii 1857 |
. . . . 5
⊢
(∃𝑎∃𝑏∃𝑐(𝑏 = 𝑎 ∧ 𝑏 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ ∃𝑎∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
25 | 19, 24 | bitri 278 |
. . . 4
⊢
(∃𝑏∃𝑎∃𝑐(𝑏 = 𝑎 ∧ 𝑏 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ ∃𝑎∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
26 | 18, 25 | bitri 278 |
. . 3
⊢
(∃𝑡∃𝑎∃𝑐(𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ ∃𝑎∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐)) |
27 | 26 | a1i 11 |
. 2
⊢ (𝑡 = 𝑏 → (∃𝑡∃𝑎∃𝑐(𝑡 = 𝑎 ∧ 𝑡 ≠ 𝑐 ∧ 𝑎 ≠ 𝑐) ↔ ∃𝑎∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐))) |
28 | 6, 13, 27 | ichcircshi 44579 |
1
⊢ [𝑎⇄𝑏]∃𝑎∃𝑏∃𝑐(𝑎 = 𝑏 ∧ 𝑎 ≠ 𝑐 ∧ 𝑏 ≠ 𝑐) |