Step | Hyp | Ref
| Expression |
1 | | eqeq1 2737 |
. . . . 5
⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩)) |
2 | 1 | anbi1d 631 |
. . . 4
⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
3 | 2 | 2exbidv 1928 |
. . 3
⊢ (𝑝 = ⟨𝑥, 𝑦⟩ → (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
4 | | eqeq1 2737 |
. . . . 5
⊢ (𝑝 = ⟨𝑖, 𝑗⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩)) |
5 | 4 | anbi1d 631 |
. . . 4
⊢ (𝑝 = ⟨𝑖, 𝑗⟩ → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
6 | 5 | 2exbidv 1928 |
. . 3
⊢ (𝑝 = ⟨𝑖, 𝑗⟩ → (∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
7 | 3, 6 | reuop 6293 |
. 2
⊢
(∃!𝑝 ∈
(𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) |
8 | | simpll 766 |
. . . . . 6
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) → 𝑥 ∈ 𝑋) |
9 | | simplr 768 |
. . . . . 6
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) → 𝑦 ∈ 𝑋) |
10 | | oppr 45740 |
. . . . . . . . . . 11
⊢ ((𝑥 ∈ V ∧ 𝑦 ∈ V) → (⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ → {𝑥, 𝑦} = {𝑎, 𝑏})) |
11 | 10 | el2v 3483 |
. . . . . . . . . 10
⊢
(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ → {𝑥, 𝑦} = {𝑎, 𝑏}) |
12 | 11 | anim1i 616 |
. . . . . . . . 9
⊢
((⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ({𝑥, 𝑦} = {𝑎, 𝑏} ∧ 𝜑)) |
13 | 12 | 2eximi 1839 |
. . . . . . . 8
⊢
(∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃𝑎∃𝑏({𝑥, 𝑦} = {𝑎, 𝑏} ∧ 𝜑)) |
14 | 13 | adantr 482 |
. . . . . . 7
⊢
((∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩)) → ∃𝑎∃𝑏({𝑥, 𝑦} = {𝑎, 𝑏} ∧ 𝜑)) |
15 | 14 | adantl 483 |
. . . . . 6
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) → ∃𝑎∃𝑏({𝑥, 𝑦} = {𝑎, 𝑏} ∧ 𝜑)) |
16 | | nfv 1918 |
. . . . . . . . . 10
⊢
Ⅎ𝑎(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) |
17 | | nfe1 2148 |
. . . . . . . . . . 11
⊢
Ⅎ𝑎∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) |
18 | | nfcv 2904 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑎𝑋 |
19 | | nfe1 2148 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑎∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) |
20 | | nfv 1918 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑎⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩ |
21 | 19, 20 | nfim 1900 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑎(∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) |
22 | 18, 21 | nfralw 3309 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑎∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) |
23 | 18, 22 | nfralw 3309 |
. . . . . . . . . . 11
⊢
Ⅎ𝑎∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) |
24 | 17, 23 | nfan 1903 |
. . . . . . . . . 10
⊢
Ⅎ𝑎(∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩)) |
25 | 16, 24 | nfan 1903 |
. . . . . . . . 9
⊢
Ⅎ𝑎((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) |
26 | | nfv 1918 |
. . . . . . . . 9
⊢
Ⅎ𝑎(𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) |
27 | 25, 26 | nfan 1903 |
. . . . . . . 8
⊢
Ⅎ𝑎(((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) |
28 | | nfv 1918 |
. . . . . . . 8
⊢
Ⅎ𝑎{𝑚, 𝑛} = {𝑥, 𝑦} |
29 | | nfv 1918 |
. . . . . . . . . . 11
⊢
Ⅎ𝑏(𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) |
30 | | nfe1 2148 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑏∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) |
31 | 30 | nfex 2318 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑏∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) |
32 | | nfcv 2904 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑏𝑋 |
33 | | nfe1 2148 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑏∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) |
34 | 33 | nfex 2318 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑏∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) |
35 | | nfv 1918 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑏⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩ |
36 | 34, 35 | nfim 1900 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑏(∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) |
37 | 32, 36 | nfralw 3309 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑏∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) |
38 | 32, 37 | nfralw 3309 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑏∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) |
39 | 31, 38 | nfan 1903 |
. . . . . . . . . . 11
⊢
Ⅎ𝑏(∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩)) |
40 | 29, 39 | nfan 1903 |
. . . . . . . . . 10
⊢
Ⅎ𝑏((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) |
41 | | nfv 1918 |
. . . . . . . . . 10
⊢
Ⅎ𝑏(𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) |
42 | 40, 41 | nfan 1903 |
. . . . . . . . 9
⊢
Ⅎ𝑏(((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) |
43 | | nfv 1918 |
. . . . . . . . 9
⊢
Ⅎ𝑏{𝑚, 𝑛} = {𝑥, 𝑦} |
44 | | vex 3479 |
. . . . . . . . . . . 12
⊢ 𝑚 ∈ V |
45 | | vex 3479 |
. . . . . . . . . . . 12
⊢ 𝑛 ∈ V |
46 | | vex 3479 |
. . . . . . . . . . . 12
⊢ 𝑎 ∈ V |
47 | | vex 3479 |
. . . . . . . . . . . 12
⊢ 𝑏 ∈ V |
48 | 44, 45, 46, 47 | preq12b 4852 |
. . . . . . . . . . 11
⊢ ({𝑚, 𝑛} = {𝑎, 𝑏} ↔ ((𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∨ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎))) |
49 | | opeq1 4874 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = 𝑚 → ⟨𝑖, 𝑗⟩ = ⟨𝑚, 𝑗⟩) |
50 | 49 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 𝑚 → (⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑚, 𝑗⟩ = ⟨𝑎, 𝑏⟩)) |
51 | 50 | anbi1d 631 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑚 → ((⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑚, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
52 | 51 | 2exbidv 1928 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑚 → (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝑚, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
53 | 49 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑚 → (⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑚, 𝑗⟩ = ⟨𝑥, 𝑦⟩)) |
54 | 52, 53 | imbi12d 345 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑚 → ((∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) ↔ (∃𝑎∃𝑏(⟨𝑚, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) |
55 | | opeq2 4875 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑛 → ⟨𝑚, 𝑗⟩ = ⟨𝑚, 𝑛⟩) |
56 | 55 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑛 → (⟨𝑚, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩)) |
57 | 56 | anbi1d 631 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑛 → ((⟨𝑚, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
58 | 57 | 2exbidv 1928 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑛 → (∃𝑎∃𝑏(⟨𝑚, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
59 | 55 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑛 → (⟨𝑚, 𝑗⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩)) |
60 | 58, 59 | imbi12d 345 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑛 → ((∃𝑎∃𝑏(⟨𝑚, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑗⟩ = ⟨𝑥, 𝑦⟩) ↔ (∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩))) |
61 | 54, 60 | rspc2v 3623 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → (∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) → (∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩))) |
62 | | pm3.22 461 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋))) |
63 | 62 | 3adant2 1132 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋))) |
64 | 63 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋))) |
65 | | eqidd 2734 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩) |
66 | | sbceq1a 3789 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑎 = 𝑚 → (𝜑 ↔ [𝑚 / 𝑎]𝜑)) |
67 | 66 | equcoms 2024 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 = 𝑎 → (𝜑 ↔ [𝑚 / 𝑎]𝜑)) |
68 | | sbceq1a 3789 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑏 = 𝑛 → ([𝑚 / 𝑎]𝜑 ↔ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑)) |
69 | 68 | equcoms 2024 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = 𝑏 → ([𝑚 / 𝑎]𝜑 ↔ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑)) |
70 | 67, 69 | sylan9bb 511 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑚 = 𝑎 ∧ 𝑛 = 𝑏) → (𝜑 ↔ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑)) |
71 | 70 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝜑 ↔ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑)) |
72 | 71 | biimpa 478 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → [𝑛 / 𝑏][𝑚 / 𝑎]𝜑) |
73 | 64, 65, 72 | jca32 517 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) ∧ (⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩ ∧ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑))) |
74 | | nfv 1918 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑎⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩ |
75 | | nfcv 2904 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑎𝑛 |
76 | | nfsbc1v 3798 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑎[𝑚 / 𝑎]𝜑 |
77 | 75, 76 | nfsbcw 3800 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑎[𝑛 / 𝑏][𝑚 / 𝑎]𝜑 |
78 | 74, 77 | nfan 1903 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑎(⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩ ∧ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑) |
79 | | nfv 1918 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑏⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩ |
80 | | nfsbc1v 3798 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑏[𝑛 / 𝑏][𝑚 / 𝑎]𝜑 |
81 | 79, 80 | nfan 1903 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩ ∧ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑) |
82 | | opeq12 4876 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑎 = 𝑚 ∧ 𝑏 = 𝑛) → ⟨𝑎, 𝑏⟩ = ⟨𝑚, 𝑛⟩) |
83 | 82 | eqeq2d 2744 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑎 = 𝑚 ∧ 𝑏 = 𝑛) → (⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩)) |
84 | 66, 68 | sylan9bb 511 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑎 = 𝑚 ∧ 𝑏 = 𝑛) → (𝜑 ↔ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑)) |
85 | 83, 84 | anbi12d 632 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑎 = 𝑚 ∧ 𝑏 = 𝑛) → ((⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩ ∧ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑))) |
86 | 85 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑎 = 𝑚 ∧ 𝑏 = 𝑛)) → ((⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩ ∧ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑))) |
87 | 78, 81, 86 | spc2ed 3592 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) → ((⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩ ∧ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑) → ∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
88 | 87 | imp 408 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) ∧ (⟨𝑚, 𝑛⟩ = ⟨𝑚, 𝑛⟩ ∧ [𝑛 / 𝑏][𝑚 / 𝑎]𝜑)) → ∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) |
89 | | pm2.27 42 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ((∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩)) |
90 | 73, 88, 89 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → ((∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩)) |
91 | | oppr 45740 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑚 ∈ V ∧ 𝑛 ∈ V) → (⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩ → {𝑚, 𝑛} = {𝑥, 𝑦})) |
92 | 91 | el2v 3483 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩ → {𝑚, 𝑛} = {𝑥, 𝑦}) |
93 | 90, 92 | syl6 35 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → ((∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩) → {𝑚, 𝑛} = {𝑥, 𝑦})) |
94 | 93 | ex 414 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝜑 → ((∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩) → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
95 | 94 | com23 86 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
96 | 95 | 3exp 1120 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → ((𝑚 = 𝑎 ∧ 𝑛 = 𝑏) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))))) |
97 | 96 | com24 95 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → ((∃𝑎∃𝑏(⟨𝑚, 𝑛⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑚, 𝑛⟩ = ⟨𝑥, 𝑦⟩) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑚 = 𝑎 ∧ 𝑛 = 𝑏) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))))) |
98 | 61, 97 | syld 47 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → (∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑚 = 𝑎 ∧ 𝑛 = 𝑏) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))))) |
99 | 98 | com13 88 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) → ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → ((𝑚 = 𝑎 ∧ 𝑛 = 𝑏) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))))) |
100 | 99 | a1d 25 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → (∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) → ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → ((𝑚 = 𝑎 ∧ 𝑛 = 𝑏) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦})))))) |
101 | 100 | imp42 428 |
. . . . . . . . . . . 12
⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) → ((𝑚 = 𝑎 ∧ 𝑛 = 𝑏) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
102 | | opeq1 4874 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 = 𝑛 → ⟨𝑖, 𝑗⟩ = ⟨𝑛, 𝑗⟩) |
103 | 102 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑖 = 𝑛 → (⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑛, 𝑗⟩ = ⟨𝑎, 𝑏⟩)) |
104 | 103 | anbi1d 631 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 = 𝑛 → ((⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑛, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
105 | 104 | 2exbidv 1928 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑛 → (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝑛, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
106 | 102 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 = 𝑛 → (⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑛, 𝑗⟩ = ⟨𝑥, 𝑦⟩)) |
107 | 105, 106 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑛 → ((∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) ↔ (∃𝑎∃𝑏(⟨𝑛, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) |
108 | | opeq2 4875 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑗 = 𝑚 → ⟨𝑛, 𝑗⟩ = ⟨𝑛, 𝑚⟩) |
109 | 108 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑗 = 𝑚 → (⟨𝑛, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩)) |
110 | 109 | anbi1d 631 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑗 = 𝑚 → ((⟨𝑛, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
111 | 110 | 2exbidv 1928 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑚 → (∃𝑎∃𝑏(⟨𝑛, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ ∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
112 | 108 | eqeq1d 2735 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑚 → (⟨𝑛, 𝑗⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩)) |
113 | 111, 112 | imbi12d 345 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑚 → ((∃𝑎∃𝑏(⟨𝑛, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑗⟩ = ⟨𝑥, 𝑦⟩) ↔ (∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩))) |
114 | 107, 113 | rspc2v 3623 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑛 ∈ 𝑋 ∧ 𝑚 ∈ 𝑋) → (∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) → (∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩))) |
115 | 114 | ancoms 460 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → (∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) → (∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩))) |
116 | | pm3.22 461 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → (𝑛 ∈ 𝑋 ∧ 𝑚 ∈ 𝑋)) |
117 | 116 | anim1ci 617 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑛 ∈ 𝑋 ∧ 𝑚 ∈ 𝑋))) |
118 | 117 | 3adant2 1132 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑛 ∈ 𝑋 ∧ 𝑚 ∈ 𝑋))) |
119 | 118 | adantr 482 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑛 ∈ 𝑋 ∧ 𝑚 ∈ 𝑋))) |
120 | | eqidd 2734 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩) |
121 | | sbceq1a 3789 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑏 = 𝑚 → (𝜑 ↔ [𝑚 / 𝑏]𝜑)) |
122 | 121 | equcoms 2024 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑚 = 𝑏 → (𝜑 ↔ [𝑚 / 𝑏]𝜑)) |
123 | | sbceq1a 3789 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑎 = 𝑛 → ([𝑚 / 𝑏]𝜑 ↔ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑)) |
124 | 123 | equcoms 2024 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑛 = 𝑎 → ([𝑚 / 𝑏]𝜑 ↔ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑)) |
125 | 122, 124 | sylan9bb 511 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑚 = 𝑏 ∧ 𝑛 = 𝑎) → (𝜑 ↔ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑)) |
126 | 125 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝜑 ↔ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑)) |
127 | 126 | biimpa 478 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → [𝑛 / 𝑎][𝑚 / 𝑏]𝜑) |
128 | 119, 120,
127 | jca32 517 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑛 ∈ 𝑋 ∧ 𝑚 ∈ 𝑋)) ∧ (⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩ ∧ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑))) |
129 | | nfv 1918 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑎⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩ |
130 | | nfsbc1v 3798 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑎[𝑛 / 𝑎][𝑚 / 𝑏]𝜑 |
131 | 129, 130 | nfan 1903 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑎(⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩ ∧ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑) |
132 | | nfv 1918 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑏⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩ |
133 | | nfcv 2904 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑏𝑛 |
134 | | nfsbc1v 3798 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
Ⅎ𝑏[𝑚 / 𝑏]𝜑 |
135 | 133, 134 | nfsbcw 3800 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑏[𝑛 / 𝑎][𝑚 / 𝑏]𝜑 |
136 | 132, 135 | nfan 1903 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩ ∧ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑) |
137 | | opeq12 4876 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑎 = 𝑛 ∧ 𝑏 = 𝑚) → ⟨𝑎, 𝑏⟩ = ⟨𝑛, 𝑚⟩) |
138 | 137 | eqeq2d 2744 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑎 = 𝑛 ∧ 𝑏 = 𝑚) → (⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩)) |
139 | 121, 123 | sylan9bbr 512 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑎 = 𝑛 ∧ 𝑏 = 𝑚) → (𝜑 ↔ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑)) |
140 | 138, 139 | anbi12d 632 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑎 = 𝑛 ∧ 𝑏 = 𝑚) → ((⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩ ∧ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑))) |
141 | 140 | adantl 483 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑎 = 𝑛 ∧ 𝑏 = 𝑚)) → ((⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ↔ (⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩ ∧ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑))) |
142 | 131, 136,
141 | spc2ed 3592 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑛 ∈ 𝑋 ∧ 𝑚 ∈ 𝑋)) → ((⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩ ∧ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑) → ∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑))) |
143 | 142 | imp 408 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (𝑛 ∈ 𝑋 ∧ 𝑚 ∈ 𝑋)) ∧ (⟨𝑛, 𝑚⟩ = ⟨𝑛, 𝑚⟩ ∧ [𝑛 / 𝑎][𝑚 / 𝑏]𝜑)) → ∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑)) |
144 | | pm2.27 42 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ((∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩)) |
145 | 128, 143,
144 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → ((∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩)) |
146 | | prcom 4737 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ {𝑛, 𝑚} = {𝑚, 𝑛} |
147 | | oppr 45740 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑛 ∈ V ∧ 𝑚 ∈ V) → (⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩ → {𝑛, 𝑚} = {𝑥, 𝑦})) |
148 | 147 | el2v 3483 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩ → {𝑛, 𝑚} = {𝑥, 𝑦}) |
149 | 146, 148 | eqtr3id 2787 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩ → {𝑚, 𝑛} = {𝑥, 𝑦}) |
150 | 145, 149 | syl6 35 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) ∧ 𝜑) → ((∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩) → {𝑚, 𝑛} = {𝑥, 𝑦})) |
151 | 150 | ex 414 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → (𝜑 → ((∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩) → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
152 | 151 | com23 86 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) ∧ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎) ∧ (𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋)) → ((∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
153 | 152 | 3exp 1120 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → ((𝑚 = 𝑏 ∧ 𝑛 = 𝑎) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))))) |
154 | 153 | com24 95 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → ((∃𝑎∃𝑏(⟨𝑛, 𝑚⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑛, 𝑚⟩ = ⟨𝑥, 𝑦⟩) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑚 = 𝑏 ∧ 𝑛 = 𝑎) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))))) |
155 | 115, 154 | syld 47 |
. . . . . . . . . . . . . . 15
⊢ ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → (∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) → ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((𝑚 = 𝑏 ∧ 𝑛 = 𝑎) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))))) |
156 | 155 | com13 88 |
. . . . . . . . . . . . . 14
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) → ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → ((𝑚 = 𝑏 ∧ 𝑛 = 𝑎) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))))) |
157 | 156 | a1d 25 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → (∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩) → ((𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋) → ((𝑚 = 𝑏 ∧ 𝑛 = 𝑎) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦})))))) |
158 | 157 | imp42 428 |
. . . . . . . . . . . 12
⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) → ((𝑚 = 𝑏 ∧ 𝑛 = 𝑎) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
159 | 101, 158 | jaod 858 |
. . . . . . . . . . 11
⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) → (((𝑚 = 𝑎 ∧ 𝑛 = 𝑏) ∨ (𝑚 = 𝑏 ∧ 𝑛 = 𝑎)) → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
160 | 48, 159 | biimtrid 241 |
. . . . . . . . . 10
⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) → ({𝑚, 𝑛} = {𝑎, 𝑏} → (𝜑 → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
161 | 160 | impd 412 |
. . . . . . . . 9
⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) → (({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑦})) |
162 | 42, 43, 161 | exlimd 2212 |
. . . . . . . 8
⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) → (∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑦})) |
163 | 27, 28, 162 | exlimd 2212 |
. . . . . . 7
⊢ ((((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) ∧ (𝑚 ∈ 𝑋 ∧ 𝑛 ∈ 𝑋)) → (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑦})) |
164 | 163 | ralrimivva 3201 |
. . . . . 6
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) → ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑦})) |
165 | | preq1 4738 |
. . . . . . . . . . 11
⊢ (𝑣 = 𝑥 → {𝑣, 𝑤} = {𝑥, 𝑤}) |
166 | 165 | eqeq1d 2735 |
. . . . . . . . . 10
⊢ (𝑣 = 𝑥 → ({𝑣, 𝑤} = {𝑎, 𝑏} ↔ {𝑥, 𝑤} = {𝑎, 𝑏})) |
167 | 166 | anbi1d 631 |
. . . . . . . . 9
⊢ (𝑣 = 𝑥 → (({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ↔ ({𝑥, 𝑤} = {𝑎, 𝑏} ∧ 𝜑))) |
168 | 167 | 2exbidv 1928 |
. . . . . . . 8
⊢ (𝑣 = 𝑥 → (∃𝑎∃𝑏({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎∃𝑏({𝑥, 𝑤} = {𝑎, 𝑏} ∧ 𝜑))) |
169 | 165 | eqeq2d 2744 |
. . . . . . . . . 10
⊢ (𝑣 = 𝑥 → ({𝑚, 𝑛} = {𝑣, 𝑤} ↔ {𝑚, 𝑛} = {𝑥, 𝑤})) |
170 | 169 | imbi2d 341 |
. . . . . . . . 9
⊢ (𝑣 = 𝑥 → ((∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑣, 𝑤}) ↔ (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑤}))) |
171 | 170 | 2ralbidv 3219 |
. . . . . . . 8
⊢ (𝑣 = 𝑥 → (∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑣, 𝑤}) ↔ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑤}))) |
172 | 168, 171 | anbi12d 632 |
. . . . . . 7
⊢ (𝑣 = 𝑥 → ((∃𝑎∃𝑏({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑣, 𝑤})) ↔ (∃𝑎∃𝑏({𝑥, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑤})))) |
173 | | preq2 4739 |
. . . . . . . . . . 11
⊢ (𝑤 = 𝑦 → {𝑥, 𝑤} = {𝑥, 𝑦}) |
174 | 173 | eqeq1d 2735 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑦 → ({𝑥, 𝑤} = {𝑎, 𝑏} ↔ {𝑥, 𝑦} = {𝑎, 𝑏})) |
175 | 174 | anbi1d 631 |
. . . . . . . . 9
⊢ (𝑤 = 𝑦 → (({𝑥, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ↔ ({𝑥, 𝑦} = {𝑎, 𝑏} ∧ 𝜑))) |
176 | 175 | 2exbidv 1928 |
. . . . . . . 8
⊢ (𝑤 = 𝑦 → (∃𝑎∃𝑏({𝑥, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎∃𝑏({𝑥, 𝑦} = {𝑎, 𝑏} ∧ 𝜑))) |
177 | 173 | eqeq2d 2744 |
. . . . . . . . . 10
⊢ (𝑤 = 𝑦 → ({𝑚, 𝑛} = {𝑥, 𝑤} ↔ {𝑚, 𝑛} = {𝑥, 𝑦})) |
178 | 177 | imbi2d 341 |
. . . . . . . . 9
⊢ (𝑤 = 𝑦 → ((∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑤}) ↔ (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
179 | 178 | 2ralbidv 3219 |
. . . . . . . 8
⊢ (𝑤 = 𝑦 → (∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑤}) ↔ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑦}))) |
180 | 176, 179 | anbi12d 632 |
. . . . . . 7
⊢ (𝑤 = 𝑦 → ((∃𝑎∃𝑏({𝑥, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑤})) ↔ (∃𝑎∃𝑏({𝑥, 𝑦} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑦})))) |
181 | 172, 180 | rspc2ev 3625 |
. . . . . 6
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋 ∧ (∃𝑎∃𝑏({𝑥, 𝑦} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑥, 𝑦}))) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (∃𝑎∃𝑏({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑣, 𝑤}))) |
182 | 8, 9, 15, 164, 181 | syl112anc 1375 |
. . . . 5
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩))) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (∃𝑎∃𝑏({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑣, 𝑤}))) |
183 | 182 | ex 414 |
. . . 4
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → ((∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩)) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (∃𝑎∃𝑏({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑣, 𝑤})))) |
184 | 183 | rexlimivv 3200 |
. . 3
⊢
(∃𝑥 ∈
𝑋 ∃𝑦 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩)) → ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (∃𝑎∃𝑏({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑣, 𝑤}))) |
185 | | eqeq1 2737 |
. . . . . 6
⊢ (𝑝 = {𝑣, 𝑤} → (𝑝 = {𝑎, 𝑏} ↔ {𝑣, 𝑤} = {𝑎, 𝑏})) |
186 | 185 | anbi1d 631 |
. . . . 5
⊢ (𝑝 = {𝑣, 𝑤} → ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ↔ ({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑))) |
187 | 186 | 2exbidv 1928 |
. . . 4
⊢ (𝑝 = {𝑣, 𝑤} → (∃𝑎∃𝑏(𝑝 = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎∃𝑏({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑))) |
188 | | eqeq1 2737 |
. . . . . 6
⊢ (𝑝 = {𝑚, 𝑛} → (𝑝 = {𝑎, 𝑏} ↔ {𝑚, 𝑛} = {𝑎, 𝑏})) |
189 | 188 | anbi1d 631 |
. . . . 5
⊢ (𝑝 = {𝑚, 𝑛} → ((𝑝 = {𝑎, 𝑏} ∧ 𝜑) ↔ ({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑))) |
190 | 189 | 2exbidv 1928 |
. . . 4
⊢ (𝑝 = {𝑚, 𝑛} → (∃𝑎∃𝑏(𝑝 = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑))) |
191 | 187, 190 | reupr 46190 |
. . 3
⊢ (𝑋 ∈ 𝑉 → (∃!𝑝 ∈ (Pairs‘𝑋)∃𝑎∃𝑏(𝑝 = {𝑎, 𝑏} ∧ 𝜑) ↔ ∃𝑣 ∈ 𝑋 ∃𝑤 ∈ 𝑋 (∃𝑎∃𝑏({𝑣, 𝑤} = {𝑎, 𝑏} ∧ 𝜑) ∧ ∀𝑚 ∈ 𝑋 ∀𝑛 ∈ 𝑋 (∃𝑎∃𝑏({𝑚, 𝑛} = {𝑎, 𝑏} ∧ 𝜑) → {𝑚, 𝑛} = {𝑣, 𝑤})))) |
192 | 184, 191 | imbitrrid 245 |
. 2
⊢ (𝑋 ∈ 𝑉 → (∃𝑥 ∈ 𝑋 ∃𝑦 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑥, 𝑦⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) ∧ ∀𝑖 ∈ 𝑋 ∀𝑗 ∈ 𝑋 (∃𝑎∃𝑏(⟨𝑖, 𝑗⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ⟨𝑖, 𝑗⟩ = ⟨𝑥, 𝑦⟩)) → ∃!𝑝 ∈ (Pairs‘𝑋)∃𝑎∃𝑏(𝑝 = {𝑎, 𝑏} ∧ 𝜑))) |
193 | 7, 192 | biimtrid 241 |
1
⊢ (𝑋 ∈ 𝑉 → (∃!𝑝 ∈ (𝑋 × 𝑋)∃𝑎∃𝑏(𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝜑) → ∃!𝑝 ∈ (Pairs‘𝑋)∃𝑎∃𝑏(𝑝 = {𝑎, 𝑏} ∧ 𝜑))) |