Proof of Theorem 2reu5
Step | Hyp | Ref
| Expression |
1 | | r19.29r 3183 |
. . . . . . . 8
⊢
((∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑥 ∈ 𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
2 | | r19.29r 3183 |
. . . . . . . . 9
⊢
((∃𝑦 ∈
𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑦 ∈ 𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
3 | 2 | reximi 3173 |
. . . . . . . 8
⊢
(∃𝑥 ∈
𝐴 (∃𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
4 | | pm3.35 803 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) |
5 | 4 | reximi 3173 |
. . . . . . . . 9
⊢
(∃𝑦 ∈
𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑦 ∈ 𝐵 (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) |
6 | 5 | reximi 3173 |
. . . . . . . 8
⊢
(∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 (𝜑 ∧ (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) |
7 | | eleq1w 2821 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)) |
8 | | eleq1w 2821 |
. . . . . . . . . . . . 13
⊢ (𝑦 = 𝑤 → (𝑦 ∈ 𝐵 ↔ 𝑤 ∈ 𝐵)) |
9 | 7, 8 | bi2anan9 639 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))) |
10 | 9 | biimpac 482 |
. . . . . . . . . . 11
⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)) |
11 | 10 | ancomd 465 |
. . . . . . . . . 10
⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)) |
12 | 11 | ex 416 |
. . . . . . . . 9
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴))) |
13 | 12 | rexlimivv 3219 |
. . . . . . . 8
⊢
(∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 (𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)) |
14 | 1, 3, 6, 13 | 4syl 19 |
. . . . . . 7
⊢
((∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴)) |
15 | 14 | ex 416 |
. . . . . 6
⊢
(∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) → (𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴))) |
16 | 15 | pm4.71rd 566 |
. . . . 5
⊢
(∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))) |
17 | | anass 472 |
. . . . 5
⊢ (((𝑤 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))) |
18 | 16, 17 | bitrdi 290 |
. . . 4
⊢
(∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ (𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))) |
19 | 18 | 2exbidv 1932 |
. . 3
⊢
(∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 𝜑 → (∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))) |
20 | 19 | pm5.32i 578 |
. 2
⊢
((∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))) |
21 | | 2reu5lem3 3684 |
. 2
⊢
((∃!𝑥 ∈
𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
22 | | df-rex 3068 |
. . . 4
⊢
(∃𝑧 ∈
𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
23 | | r19.42v 3275 |
. . . . . 6
⊢
(∃𝑤 ∈
𝐵 (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |
24 | | df-rex 3068 |
. . . . . 6
⊢
(∃𝑤 ∈
𝐵 (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))) |
25 | 23, 24 | bitr3i 280 |
. . . . 5
⊢ ((𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ ∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))) |
26 | 25 | exbii 1855 |
. . . 4
⊢
(∃𝑧(𝑧 ∈ 𝐴 ∧ ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))) |
27 | 22, 26 | bitri 278 |
. . 3
⊢
(∃𝑧 ∈
𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)) ↔ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))))) |
28 | 27 | anbi2i 626 |
. 2
⊢
((∃𝑥 ∈
𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤))) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧∃𝑤(𝑤 ∈ 𝐵 ∧ (𝑧 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))))) |
29 | 20, 21, 28 | 3bitr4i 306 |
1
⊢
((∃!𝑥 ∈
𝐴 ∃!𝑦 ∈ 𝐵 𝜑 ∧ ∀𝑥 ∈ 𝐴 ∃*𝑦 ∈ 𝐵 𝜑) ↔ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ∧ ∃𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐵 ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜑 → (𝑥 = 𝑧 ∧ 𝑦 = 𝑤)))) |