| Step | Hyp | Ref
| Expression |
| 1 | | c0ex 11255 |
. . . 4
⊢ 0 ∈
V |
| 2 | | zindbi.4 |
. . . 4
⊢ (𝑥 = 0 → (𝜑 ↔ 𝜃)) |
| 3 | 1, 2 | sbcie 3830 |
. . 3
⊢
([0 / 𝑥]𝜑 ↔ 𝜃) |
| 4 | | 0z 12624 |
. . . . 5
⊢ 0 ∈
ℤ |
| 5 | | eleq1 2829 |
. . . . . . . . . 10
⊢ (𝑦 = 0 → (𝑦 ∈ ℤ ↔ 0 ∈
ℤ)) |
| 6 | | breq1 5146 |
. . . . . . . . . 10
⊢ (𝑦 = 0 → (𝑦 ≤ 𝑏 ↔ 0 ≤ 𝑏)) |
| 7 | 5, 6 | 3anbi13d 1440 |
. . . . . . . . 9
⊢ (𝑦 = 0 → ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) ↔ (0 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 0 ≤
𝑏))) |
| 8 | | dfsbcq 3790 |
. . . . . . . . . 10
⊢ (𝑦 = 0 → ([𝑦 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑)) |
| 9 | 8 | bibi1d 343 |
. . . . . . . . 9
⊢ (𝑦 = 0 → (([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([0 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑))) |
| 10 | 7, 9 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑦 = 0 → (((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) ↔ ((0 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 0 ≤
𝑏) → ([0 /
𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)))) |
| 11 | | eleq1 2829 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐴 → (𝑏 ∈ ℤ ↔ 𝐴 ∈ ℤ)) |
| 12 | | breq2 5147 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐴 → (0 ≤ 𝑏 ↔ 0 ≤ 𝐴)) |
| 13 | 11, 12 | 3anbi23d 1441 |
. . . . . . . . 9
⊢ (𝑏 = 𝐴 → ((0 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 0 ≤
𝑏) ↔ (0 ∈ ℤ
∧ 𝐴 ∈ ℤ
∧ 0 ≤ 𝐴))) |
| 14 | | dfsbcq 3790 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐴 → ([𝑏 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
| 15 | 14 | bibi2d 342 |
. . . . . . . . 9
⊢ (𝑏 = 𝐴 → (([0 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))) |
| 16 | 13, 15 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑏 = 𝐴 → (((0 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 0 ≤
𝑏) → ([0 /
𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) ↔ ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤
𝐴) → ([0 /
𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)))) |
| 17 | | dfsbcq 3790 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑦 → ([𝑎 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
| 18 | 17 | bibi2d 342 |
. . . . . . . . 9
⊢ (𝑎 = 𝑦 → (([𝑦 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))) |
| 19 | | dfsbcq 3790 |
. . . . . . . . . 10
⊢ (𝑎 = 𝑏 → ([𝑎 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) |
| 20 | 19 | bibi2d 342 |
. . . . . . . . 9
⊢ (𝑎 = 𝑏 → (([𝑦 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑))) |
| 21 | | dfsbcq 3790 |
. . . . . . . . . 10
⊢ (𝑎 = (𝑏 + 1) → ([𝑎 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑)) |
| 22 | 21 | bibi2d 342 |
. . . . . . . . 9
⊢ (𝑎 = (𝑏 + 1) → (([𝑦 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑))) |
| 23 | | biidd 262 |
. . . . . . . . 9
⊢ (𝑦 ∈ ℤ →
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)) |
| 24 | | vex 3484 |
. . . . . . . . . . . . . . . 16
⊢ 𝑦 ∈ V |
| 25 | | zindbi.2 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
| 26 | 24, 25 | sbcie 3830 |
. . . . . . . . . . . . . . 15
⊢
([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
| 27 | | dfsbcq 3790 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑏 → ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) |
| 28 | 26, 27 | bitr3id 285 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑏 → (𝜓 ↔ [𝑏 / 𝑥]𝜑)) |
| 29 | | ovex 7464 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 + 1) ∈ V |
| 30 | | zindbi.3 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑦 + 1) → (𝜑 ↔ 𝜒)) |
| 31 | 29, 30 | sbcie 3830 |
. . . . . . . . . . . . . . 15
⊢
([(𝑦 + 1) /
𝑥]𝜑 ↔ 𝜒) |
| 32 | | oveq1 7438 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = 𝑏 → (𝑦 + 1) = (𝑏 + 1)) |
| 33 | 32 | sbceq1d 3793 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑏 → ([(𝑦 + 1) / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑)) |
| 34 | 31, 33 | bitr3id 285 |
. . . . . . . . . . . . . 14
⊢ (𝑦 = 𝑏 → (𝜒 ↔ [(𝑏 + 1) / 𝑥]𝜑)) |
| 35 | 28, 34 | bibi12d 345 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑏 → ((𝜓 ↔ 𝜒) ↔ ([𝑏 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑))) |
| 36 | | zindbi.1 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ ℤ → (𝜓 ↔ 𝜒)) |
| 37 | 35, 36 | vtoclga 3577 |
. . . . . . . . . . . 12
⊢ (𝑏 ∈ ℤ →
([𝑏 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑)) |
| 38 | 37 | 3ad2ant2 1135 |
. . . . . . . . . . 11
⊢ ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → ([𝑏 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑)) |
| 39 | 38 | bibi2d 342 |
. . . . . . . . . 10
⊢ ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → (([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([𝑦 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑))) |
| 40 | 39 | biimpd 229 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → (([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) → ([𝑦 / 𝑥]𝜑 ↔ [(𝑏 + 1) / 𝑥]𝜑))) |
| 41 | 18, 20, 22, 20, 23, 40 | uzind 12710 |
. . . . . . . 8
⊢ ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) |
| 42 | 10, 16, 41 | vtocl2g 3574 |
. . . . . . 7
⊢ ((0
∈ ℤ ∧ 𝐴
∈ ℤ) → ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤ 𝐴) → ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))) |
| 43 | 42 | 3adant3 1133 |
. . . . . 6
⊢ ((0
∈ ℤ ∧ 𝐴
∈ ℤ ∧ 0 ≤ 𝐴) → ((0 ∈ ℤ ∧ 𝐴 ∈ ℤ ∧ 0 ≤
𝐴) → ([0 /
𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))) |
| 44 | 43 | pm2.43i 52 |
. . . . 5
⊢ ((0
∈ ℤ ∧ 𝐴
∈ ℤ ∧ 0 ≤ 𝐴) → ([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
| 45 | 4, 44 | mp3an1 1450 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 0 ≤
𝐴) → ([0 /
𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
| 46 | | eleq1 2829 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (𝑦 ∈ ℤ ↔ 𝐴 ∈ ℤ)) |
| 47 | | breq1 5146 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → (𝑦 ≤ 𝑏 ↔ 𝐴 ≤ 𝑏)) |
| 48 | 46, 47 | 3anbi13d 1440 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → ((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) ↔ (𝐴 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝐴 ≤ 𝑏))) |
| 49 | | dfsbcq 3790 |
. . . . . . . . . . 11
⊢ (𝑦 = 𝐴 → ([𝑦 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
| 50 | 49 | bibi1d 343 |
. . . . . . . . . 10
⊢ (𝑦 = 𝐴 → (([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑))) |
| 51 | 48, 50 | imbi12d 344 |
. . . . . . . . 9
⊢ (𝑦 = 𝐴 → (((𝑦 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝑦 ≤ 𝑏) → ([𝑦 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) ↔ ((𝐴 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝐴 ≤ 𝑏) → ([𝐴 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)))) |
| 52 | | eleq1 2829 |
. . . . . . . . . . 11
⊢ (𝑏 = 0 → (𝑏 ∈ ℤ ↔ 0 ∈
ℤ)) |
| 53 | | breq2 5147 |
. . . . . . . . . . 11
⊢ (𝑏 = 0 → (𝐴 ≤ 𝑏 ↔ 𝐴 ≤ 0)) |
| 54 | 52, 53 | 3anbi23d 1441 |
. . . . . . . . . 10
⊢ (𝑏 = 0 → ((𝐴 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝐴 ≤ 𝑏) ↔ (𝐴 ∈ ℤ ∧ 0 ∈ ℤ ∧
𝐴 ≤
0))) |
| 55 | | dfsbcq 3790 |
. . . . . . . . . . 11
⊢ (𝑏 = 0 → ([𝑏 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑)) |
| 56 | 55 | bibi2d 342 |
. . . . . . . . . 10
⊢ (𝑏 = 0 → (([𝐴 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑) ↔ ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑))) |
| 57 | 54, 56 | imbi12d 344 |
. . . . . . . . 9
⊢ (𝑏 = 0 → (((𝐴 ∈ ℤ ∧ 𝑏 ∈ ℤ ∧ 𝐴 ≤ 𝑏) → ([𝐴 / 𝑥]𝜑 ↔ [𝑏 / 𝑥]𝜑)) ↔ ((𝐴 ∈ ℤ ∧ 0 ∈ ℤ ∧
𝐴 ≤ 0) →
([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑)))) |
| 58 | 51, 57, 41 | vtocl2g 3574 |
. . . . . . . 8
⊢ ((𝐴 ∈ ℤ ∧ 0 ∈
ℤ) → ((𝐴 ∈
ℤ ∧ 0 ∈ ℤ ∧ 𝐴 ≤ 0) → ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑))) |
| 59 | 58 | 3adant3 1133 |
. . . . . . 7
⊢ ((𝐴 ∈ ℤ ∧ 0 ∈
ℤ ∧ 𝐴 ≤ 0)
→ ((𝐴 ∈ ℤ
∧ 0 ∈ ℤ ∧ 𝐴 ≤ 0) → ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑))) |
| 60 | 59 | pm2.43i 52 |
. . . . . 6
⊢ ((𝐴 ∈ ℤ ∧ 0 ∈
ℤ ∧ 𝐴 ≤ 0)
→ ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑)) |
| 61 | 4, 60 | mp3an2 1451 |
. . . . 5
⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ≤ 0) → ([𝐴 / 𝑥]𝜑 ↔ [0 / 𝑥]𝜑)) |
| 62 | 61 | bicomd 223 |
. . . 4
⊢ ((𝐴 ∈ ℤ ∧ 𝐴 ≤ 0) → ([0 /
𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
| 63 | | 0re 11263 |
. . . . 5
⊢ 0 ∈
ℝ |
| 64 | | zre 12617 |
. . . . 5
⊢ (𝐴 ∈ ℤ → 𝐴 ∈
ℝ) |
| 65 | | letric 11361 |
. . . . 5
⊢ ((0
∈ ℝ ∧ 𝐴
∈ ℝ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0)) |
| 66 | 63, 64, 65 | sylancr 587 |
. . . 4
⊢ (𝐴 ∈ ℤ → (0 ≤
𝐴 ∨ 𝐴 ≤ 0)) |
| 67 | 45, 62, 66 | mpjaodan 961 |
. . 3
⊢ (𝐴 ∈ ℤ →
([0 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑)) |
| 68 | 3, 67 | bitr3id 285 |
. 2
⊢ (𝐴 ∈ ℤ → (𝜃 ↔ [𝐴 / 𝑥]𝜑)) |
| 69 | | zindbi.5 |
. . 3
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜏)) |
| 70 | 69 | sbcieg 3828 |
. 2
⊢ (𝐴 ∈ ℤ →
([𝐴 / 𝑥]𝜑 ↔ 𝜏)) |
| 71 | 68, 70 | bitrd 279 |
1
⊢ (𝐴 ∈ ℤ → (𝜃 ↔ 𝜏)) |