Step | Hyp | Ref
| Expression |
1 | | opex 5426 |
. . 3
⊢
⟨𝐴, 𝐵⟩ ∈ V |
2 | | opex 5426 |
. . 3
⊢
⟨𝐶, 𝐷⟩ ∈ V |
3 | | eqeq1 2741 |
. . . . . . 7
⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩)) |
4 | 3 | 3anbi1d 1441 |
. . . . . 6
⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → ((𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
5 | 4 | rexbidv 3176 |
. . . . 5
⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
6 | 5 | 2rexbidv 3214 |
. . . 4
⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
7 | 6 | 2rexbidv 3214 |
. . 3
⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
8 | | eqeq1 2741 |
. . . . . . 7
⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩)) |
9 | 8 | 3anbi2d 1442 |
. . . . . 6
⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
10 | 9 | rexbidv 3176 |
. . . . 5
⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
11 | 10 | 2rexbidv 3214 |
. . . 4
⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
12 | 11 | 2rexbidv 3214 |
. . 3
⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
13 | | br4.6 |
. . 3
⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)} |
14 | 1, 2, 7, 12, 13 | brab 5505 |
. 2
⊢
(⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)) |
15 | | vex 3452 |
. . . . . . . . . . . 12
⊢ 𝑎 ∈ V |
16 | | vex 3452 |
. . . . . . . . . . . 12
⊢ 𝑏 ∈ V |
17 | 15, 16 | opth 5438 |
. . . . . . . . . . 11
⊢
(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ ↔ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) |
18 | | br4.1 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓)) |
19 | | br4.2 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒)) |
20 | 18, 19 | sylan9bb 511 |
. . . . . . . . . . 11
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (𝜑 ↔ 𝜒)) |
21 | 17, 20 | sylbi 216 |
. . . . . . . . . 10
⊢
(⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝐵⟩ → (𝜑 ↔ 𝜒)) |
22 | 21 | eqcoms 2745 |
. . . . . . . . 9
⊢
(⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ → (𝜑 ↔ 𝜒)) |
23 | | vex 3452 |
. . . . . . . . . . . 12
⊢ 𝑐 ∈ V |
24 | | vex 3452 |
. . . . . . . . . . . 12
⊢ 𝑑 ∈ V |
25 | 23, 24 | opth 5438 |
. . . . . . . . . . 11
⊢
(⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ ↔ (𝑐 = 𝐶 ∧ 𝑑 = 𝐷)) |
26 | | br4.3 |
. . . . . . . . . . . 12
⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃)) |
27 | | br4.4 |
. . . . . . . . . . . 12
⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏)) |
28 | 26, 27 | sylan9bb 511 |
. . . . . . . . . . 11
⊢ ((𝑐 = 𝐶 ∧ 𝑑 = 𝐷) → (𝜒 ↔ 𝜏)) |
29 | 25, 28 | sylbi 216 |
. . . . . . . . . 10
⊢
(⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝐷⟩ → (𝜒 ↔ 𝜏)) |
30 | 29 | eqcoms 2745 |
. . . . . . . . 9
⊢
(⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ → (𝜒 ↔ 𝜏)) |
31 | 22, 30 | sylan9bb 511 |
. . . . . . . 8
⊢
((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩) → (𝜑 ↔ 𝜏)) |
32 | 31 | biimp3a 1470 |
. . . . . . 7
⊢
((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏) |
33 | 32 | a1i 11 |
. . . . . 6
⊢
(((((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃)) ∧ (𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃)) ∧ 𝑑 ∈ 𝑃) → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏)) |
34 | 33 | rexlimdva 3153 |
. . . . 5
⊢ ((((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃)) ∧ (𝑏 ∈ 𝑃 ∧ 𝑐 ∈ 𝑃)) → (∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏)) |
35 | 34 | rexlimdvva 3206 |
. . . 4
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑎 ∈ 𝑃)) → (∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏)) |
36 | 35 | rexlimdvva 3206 |
. . 3
⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) → 𝜏)) |
37 | | simpl1 1192 |
. . . . 5
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → 𝑋 ∈ 𝑆) |
38 | | simpl2l 1227 |
. . . . . 6
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → 𝐴 ∈ 𝑄) |
39 | | simpl2r 1228 |
. . . . . 6
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → 𝐵 ∈ 𝑄) |
40 | | simpl3l 1229 |
. . . . . . 7
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → 𝐶 ∈ 𝑄) |
41 | | simpl3r 1230 |
. . . . . . 7
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → 𝐷 ∈ 𝑄) |
42 | | eqidd 2738 |
. . . . . . 7
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩) |
43 | | eqidd 2738 |
. . . . . . 7
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩) |
44 | | simpr 486 |
. . . . . . 7
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → 𝜏) |
45 | | opeq1 4835 |
. . . . . . . . . 10
⊢ (𝑐 = 𝐶 → ⟨𝑐, 𝑑⟩ = ⟨𝐶, 𝑑⟩) |
46 | 45 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (𝑐 = 𝐶 → (⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩)) |
47 | 46, 26 | 3anbi23d 1440 |
. . . . . . . 8
⊢ (𝑐 = 𝐶 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩ ∧ 𝜃))) |
48 | | opeq2 4836 |
. . . . . . . . . 10
⊢ (𝑑 = 𝐷 → ⟨𝐶, 𝑑⟩ = ⟨𝐶, 𝐷⟩) |
49 | 48 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (𝑑 = 𝐷 → (⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩ ↔ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩)) |
50 | 49, 27 | 3anbi23d 1440 |
. . . . . . . 8
⊢ (𝑑 = 𝐷 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝑑⟩ ∧ 𝜃) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝜏))) |
51 | 47, 50 | rspc2ev 3595 |
. . . . . . 7
⊢ ((𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝐶, 𝐷⟩ ∧ 𝜏)) → ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒)) |
52 | 40, 41, 42, 43, 44, 51 | syl113anc 1383 |
. . . . . 6
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒)) |
53 | | opeq1 4835 |
. . . . . . . . . 10
⊢ (𝑎 = 𝐴 → ⟨𝑎, 𝑏⟩ = ⟨𝐴, 𝑏⟩) |
54 | 53 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (𝑎 = 𝐴 → (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩)) |
55 | 54, 18 | 3anbi13d 1439 |
. . . . . . . 8
⊢ (𝑎 = 𝐴 → ((⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓))) |
56 | 55 | 2rexbidv 3214 |
. . . . . . 7
⊢ (𝑎 = 𝐴 → (∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓))) |
57 | | opeq2 4836 |
. . . . . . . . . 10
⊢ (𝑏 = 𝐵 → ⟨𝐴, 𝑏⟩ = ⟨𝐴, 𝐵⟩) |
58 | 57 | eqeq2d 2748 |
. . . . . . . . 9
⊢ (𝑏 = 𝐵 → (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩)) |
59 | 58, 19 | 3anbi13d 1439 |
. . . . . . . 8
⊢ (𝑏 = 𝐵 → ((⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓) ↔ (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒))) |
60 | 59 | 2rexbidv 3214 |
. . . . . . 7
⊢ (𝑏 = 𝐵 → (∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜓) ↔ ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒))) |
61 | 56, 60 | rspc2ev 3595 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜒)) → ∃𝑎 ∈ 𝑄 ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)) |
62 | 38, 39, 52, 61 | syl3anc 1372 |
. . . . 5
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → ∃𝑎 ∈ 𝑄 ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)) |
63 | | br4.5 |
. . . . . . 7
⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄) |
64 | 63 | rexeqdv 3317 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → (∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
65 | 63, 64 | rexeqbidv 3323 |
. . . . . . . 8
⊢ (𝑥 = 𝑋 → (∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
66 | 63, 65 | rexeqbidv 3323 |
. . . . . . 7
⊢ (𝑥 = 𝑋 → (∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
67 | 63, 66 | rexeqbidv 3323 |
. . . . . 6
⊢ (𝑥 = 𝑋 → (∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ ∃𝑎 ∈ 𝑄 ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
68 | 67 | rspcev 3584 |
. . . . 5
⊢ ((𝑋 ∈ 𝑆 ∧ ∃𝑎 ∈ 𝑄 ∃𝑏 ∈ 𝑄 ∃𝑐 ∈ 𝑄 ∃𝑑 ∈ 𝑄 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)) → ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)) |
69 | 37, 62, 68 | syl2anc 585 |
. . . 4
⊢ (((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) ∧ 𝜏) → ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑)) |
70 | 69 | ex 414 |
. . 3
⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (𝜏 → ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑))) |
71 | 36, 70 | impbid 211 |
. 2
⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (⟨𝐴, 𝐵⟩ = ⟨𝑎, 𝑏⟩ ∧ ⟨𝐶, 𝐷⟩ = ⟨𝑐, 𝑑⟩ ∧ 𝜑) ↔ 𝜏)) |
72 | 14, 71 | bitrid 283 |
1
⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜏)) |