| Step | Hyp | Ref
| Expression |
| 1 | | raleq 3323 |
. . . 4
⊢ (𝑥 = ∅ → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ ∅ 𝜑)) |
| 2 | 1 | rexralbidv 3223 |
. . 3
⊢ (𝑥 = ∅ → (∃𝑗 ∈ ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑)) |
| 3 | | raleq 3323 |
. . 3
⊢ (𝑥 = ∅ → (∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 4 | 2, 3 | bibi12d 345 |
. 2
⊢ (𝑥 = ∅ → ((∃𝑗 ∈ ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))) |
| 5 | | raleq 3323 |
. . . 4
⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑦 𝜑)) |
| 6 | 5 | rexralbidv 3223 |
. . 3
⊢ (𝑥 = 𝑦 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑)) |
| 7 | | raleq 3323 |
. . 3
⊢ (𝑥 = 𝑦 → (∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 8 | 6, 7 | bibi12d 345 |
. 2
⊢ (𝑥 = 𝑦 → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))) |
| 9 | | raleq 3323 |
. . . 4
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑)) |
| 10 | 9 | rexralbidv 3223 |
. . 3
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑)) |
| 11 | | raleq 3323 |
. . 3
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → (∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 12 | 10, 11 | bibi12d 345 |
. 2
⊢ (𝑥 = (𝑦 ∪ {𝑧}) → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))) |
| 13 | | raleq 3323 |
. . . 4
⊢ (𝑥 = 𝐴 → (∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝐴 𝜑)) |
| 14 | 13 | rexralbidv 3223 |
. . 3
⊢ (𝑥 = 𝐴 → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑)) |
| 15 | | raleq 3323 |
. . 3
⊢ (𝑥 = 𝐴 → (∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∀𝑛 ∈ 𝐴 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 16 | 14, 15 | bibi12d 345 |
. 2
⊢ (𝑥 = 𝐴 → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑥 𝜑 ↔ ∀𝑛 ∈ 𝑥 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑 ↔ ∀𝑛 ∈ 𝐴 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))) |
| 17 | | 0z 12624 |
. . . . 5
⊢ 0 ∈
ℤ |
| 18 | 17 | ne0ii 4344 |
. . . 4
⊢ ℤ
≠ ∅ |
| 19 | | ral0 4513 |
. . . . 5
⊢
∀𝑛 ∈
∅ 𝜑 |
| 20 | 19 | rgen2w 3066 |
. . . 4
⊢
∀𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑 |
| 21 | | r19.2z 4495 |
. . . 4
⊢ ((ℤ
≠ ∅ ∧ ∀𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑) → ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑) |
| 22 | 18, 20, 21 | mp2an 692 |
. . 3
⊢
∃𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑 |
| 23 | | ral0 4513 |
. . 3
⊢
∀𝑛 ∈
∅ ∃𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)𝜑 |
| 24 | 22, 23 | 2th 264 |
. 2
⊢
(∃𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ ∅ 𝜑 ↔ ∀𝑛 ∈ ∅ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) |
| 25 | | anbi1 633 |
. . . 4
⊢
((∃𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → ((∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))) |
| 26 | | rexanuz 15384 |
. . . . 5
⊢
(∃𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)(∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑)) |
| 27 | | ralunb 4197 |
. . . . . . 7
⊢
(∀𝑛 ∈
(𝑦 ∪ {𝑧})𝜑 ↔ (∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑)) |
| 28 | 27 | ralbii 3093 |
. . . . . 6
⊢
(∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑘 ∈ (ℤ≥‘𝑗)(∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑)) |
| 29 | 28 | rexbii 3094 |
. . . . 5
⊢
(∃𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)(∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}𝜑)) |
| 30 | | ralsnsg 4670 |
. . . . . . . 8
⊢ (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ [𝑧 / 𝑛]∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 31 | | sbcrex 3875 |
. . . . . . . . 9
⊢
([𝑧 / 𝑛]∃𝑗 ∈ ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) |
| 32 | | ralcom 3289 |
. . . . . . . . . . 11
⊢
(∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) |
| 33 | | ralsnsg 4670 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ [𝑧 / 𝑛]∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 34 | 32, 33 | bitrid 283 |
. . . . . . . . . 10
⊢ (𝑧 ∈ V → (∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ [𝑧 / 𝑛]∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 35 | 34 | rexbidv 3179 |
. . . . . . . . 9
⊢ (𝑧 ∈ V → (∃𝑗 ∈ ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑 ↔ ∃𝑗 ∈ ℤ [𝑧 / 𝑛]∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 36 | 31, 35 | bitr4id 290 |
. . . . . . . 8
⊢ (𝑧 ∈ V → ([𝑧 / 𝑛]∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑)) |
| 37 | 30, 36 | bitrd 279 |
. . . . . . 7
⊢ (𝑧 ∈ V → (∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑)) |
| 38 | 37 | elv 3485 |
. . . . . 6
⊢
(∀𝑛 ∈
{𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)𝜑 ↔ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑) |
| 39 | 38 | anbi2i 623 |
. . . . 5
⊢
((∃𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ {𝑧}𝜑)) |
| 40 | 26, 29, 39 | 3bitr4i 303 |
. . . 4
⊢
(∃𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 41 | | ralunb 4197 |
. . . 4
⊢
(∀𝑛 ∈
(𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ↔ (∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑 ∧ ∀𝑛 ∈ {𝑧}∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 42 | 25, 40, 41 | 3bitr4g 314 |
. . 3
⊢
((∃𝑗 ∈
ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |
| 43 | 42 | a1i 11 |
. 2
⊢ (𝑦 ∈ Fin →
((∃𝑗 ∈ ℤ
∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ 𝑦 𝜑 ↔ ∀𝑛 ∈ 𝑦 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑) → (∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)∀𝑛 ∈ (𝑦 ∪ {𝑧})𝜑 ↔ ∀𝑛 ∈ (𝑦 ∪ {𝑧})∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑))) |
| 44 | 4, 8, 12, 16, 24, 43 | findcard2 9204 |
1
⊢ (𝐴 ∈ Fin → (∃𝑗 ∈ ℤ ∀𝑘 ∈
(ℤ≥‘𝑗)∀𝑛 ∈ 𝐴 𝜑 ↔ ∀𝑛 ∈ 𝐴 ∃𝑗 ∈ ℤ ∀𝑘 ∈ (ℤ≥‘𝑗)𝜑)) |