Proof of Theorem reusv3
Step | Hyp | Ref
| Expression |
1 | | reusv3.1 |
. . . . 5
⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜓)) |
2 | | reusv3.2 |
. . . . . 6
⊢ (𝑦 = 𝑧 → 𝐶 = 𝐷) |
3 | 2 | eleq1d 2239 |
. . . . 5
⊢ (𝑦 = 𝑧 → (𝐶 ∈ 𝐴 ↔ 𝐷 ∈ 𝐴)) |
4 | 1, 3 | anbi12d 470 |
. . . 4
⊢ (𝑦 = 𝑧 → ((𝜑 ∧ 𝐶 ∈ 𝐴) ↔ (𝜓 ∧ 𝐷 ∈ 𝐴))) |
5 | 4 | cbvrexv 2697 |
. . 3
⊢
(∃𝑦 ∈
𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) ↔ ∃𝑧 ∈ 𝐵 (𝜓 ∧ 𝐷 ∈ 𝐴)) |
6 | | nfra2xy 2512 |
. . . . 5
⊢
Ⅎ𝑧∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) |
7 | | nfv 1521 |
. . . . 5
⊢
Ⅎ𝑧∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) |
8 | 6, 7 | nfim 1565 |
. . . 4
⊢
Ⅎ𝑧(∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)) |
9 | | risset 2498 |
. . . . . 6
⊢ (𝐷 ∈ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑥 = 𝐷) |
10 | | ralcom 2633 |
. . . . . . . . . . . . . 14
⊢
(∀𝑦 ∈
𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) |
11 | | impexp 261 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜑 → (𝜓 → 𝐶 = 𝐷))) |
12 | | bi2.04 247 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 → (𝜓 → 𝐶 = 𝐷)) ↔ (𝜓 → (𝜑 → 𝐶 = 𝐷))) |
13 | 11, 12 | bitri 183 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → (𝜑 → 𝐶 = 𝐷))) |
14 | 13 | ralbii 2476 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑦 ∈
𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (𝜑 → 𝐶 = 𝐷))) |
15 | | r19.21v 2547 |
. . . . . . . . . . . . . . . 16
⊢
(∀𝑦 ∈
𝐵 (𝜓 → (𝜑 → 𝐶 = 𝐷)) ↔ (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))) |
16 | 14, 15 | bitri 183 |
. . . . . . . . . . . . . . 15
⊢
(∀𝑦 ∈
𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))) |
17 | 16 | ralbii 2476 |
. . . . . . . . . . . . . 14
⊢
(∀𝑧 ∈
𝐵 ∀𝑦 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))) |
18 | 10, 17 | bitri 183 |
. . . . . . . . . . . . 13
⊢
(∀𝑦 ∈
𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∀𝑧 ∈ 𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))) |
19 | | rsp 2517 |
. . . . . . . . . . . . 13
⊢
(∀𝑧 ∈
𝐵 (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)) → (𝑧 ∈ 𝐵 → (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))) |
20 | 18, 19 | sylbi 120 |
. . . . . . . . . . . 12
⊢
(∀𝑦 ∈
𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → (𝑧 ∈ 𝐵 → (𝜓 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))) |
21 | 20 | com3l 81 |
. . . . . . . . . . 11
⊢ (𝑧 ∈ 𝐵 → (𝜓 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)))) |
22 | 21 | imp31 254 |
. . . . . . . . . 10
⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷)) |
23 | | eqeq1 2177 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝐷 → (𝑥 = 𝐶 ↔ 𝐷 = 𝐶)) |
24 | | eqcom 2172 |
. . . . . . . . . . . . 13
⊢ (𝐷 = 𝐶 ↔ 𝐶 = 𝐷) |
25 | 23, 24 | bitrdi 195 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐷 → (𝑥 = 𝐶 ↔ 𝐶 = 𝐷)) |
26 | 25 | imbi2d 229 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐷 → ((𝜑 → 𝑥 = 𝐶) ↔ (𝜑 → 𝐶 = 𝐷))) |
27 | 26 | ralbidv 2470 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐷 → (∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜑 → 𝐶 = 𝐷))) |
28 | 22, 27 | syl5ibrcom 156 |
. . . . . . . . 9
⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → (𝑥 = 𝐷 → ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
29 | 28 | reximdv 2571 |
. . . . . . . 8
⊢ (((𝑧 ∈ 𝐵 ∧ 𝜓) ∧ ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) → (∃𝑥 ∈ 𝐴 𝑥 = 𝐷 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
30 | 29 | ex 114 |
. . . . . . 7
⊢ ((𝑧 ∈ 𝐵 ∧ 𝜓) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → (∃𝑥 ∈ 𝐴 𝑥 = 𝐷 → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))) |
31 | 30 | com23 78 |
. . . . . 6
⊢ ((𝑧 ∈ 𝐵 ∧ 𝜓) → (∃𝑥 ∈ 𝐴 𝑥 = 𝐷 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))) |
32 | 9, 31 | syl5bi 151 |
. . . . 5
⊢ ((𝑧 ∈ 𝐵 ∧ 𝜓) → (𝐷 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))) |
33 | 32 | expimpd 361 |
. . . 4
⊢ (𝑧 ∈ 𝐵 → ((𝜓 ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶)))) |
34 | 8, 33 | rexlimi 2580 |
. . 3
⊢
(∃𝑧 ∈
𝐵 (𝜓 ∧ 𝐷 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
35 | 5, 34 | sylbi 120 |
. 2
⊢
(∃𝑦 ∈
𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) → ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |
36 | 1, 2 | reusv3i 4442 |
. 2
⊢
(∃𝑥 ∈
𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶) → ∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷)) |
37 | 35, 36 | impbid1 141 |
1
⊢
(∃𝑦 ∈
𝐵 (𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑦 ∈ 𝐵 ∀𝑧 ∈ 𝐵 ((𝜑 ∧ 𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → 𝑥 = 𝐶))) |