Proof of Theorem brecop
Step | Hyp | Ref
| Expression |
1 | | brecop.1 |
. . . 4
⊢ ∼ ∈
V |
2 | | brecop.4 |
. . . 4
⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ ) |
3 | 1, 2 | ecopqsi 6535 |
. . 3
⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) → [〈𝐴, 𝐵〉] ∼ ∈ 𝐻) |
4 | 1, 2 | ecopqsi 6535 |
. . 3
⊢ ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → [〈𝐶, 𝐷〉] ∼ ∈ 𝐻) |
5 | | df-br 3966 |
. . . . 5
⊢
([〈𝐴, 𝐵〉] ∼ ≤ [〈𝐶, 𝐷〉] ∼ ↔
〈[〈𝐴, 𝐵〉] ∼ , [〈𝐶, 𝐷〉] ∼ 〉 ∈ ≤
) |
6 | | brecop.5 |
. . . . . 6
⊢ ≤ =
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))} |
7 | 6 | eleq2i 2224 |
. . . . 5
⊢
(〈[〈𝐴,
𝐵〉] ∼ , [〈𝐶, 𝐷〉] ∼ 〉 ∈ ≤ ↔
〈[〈𝐴, 𝐵〉] ∼ , [〈𝐶, 𝐷〉] ∼ 〉 ∈
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))}) |
8 | 5, 7 | bitri 183 |
. . . 4
⊢
([〈𝐴, 𝐵〉] ∼ ≤ [〈𝐶, 𝐷〉] ∼ ↔
〈[〈𝐴, 𝐵〉] ∼ , [〈𝐶, 𝐷〉] ∼ 〉 ∈
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))}) |
9 | | eqeq1 2164 |
. . . . . . . 8
⊢ (𝑥 = [〈𝐴, 𝐵〉] ∼ → (𝑥 = [〈𝑧, 𝑤〉] ∼ ↔ [〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ )) |
10 | 9 | anbi1d 461 |
. . . . . . 7
⊢ (𝑥 = [〈𝐴, 𝐵〉] ∼ → ((𝑥 = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ↔
([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼
))) |
11 | 10 | anbi1d 461 |
. . . . . 6
⊢ (𝑥 = [〈𝐴, 𝐵〉] ∼ → (((𝑥 = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑) ↔ (([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))) |
12 | 11 | 4exbidv 1850 |
. . . . 5
⊢ (𝑥 = [〈𝐴, 𝐵〉] ∼ →
(∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))) |
13 | | eqeq1 2164 |
. . . . . . . 8
⊢ (𝑦 = [〈𝐶, 𝐷〉] ∼ → (𝑦 = [〈𝑣, 𝑢〉] ∼ ↔ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ )) |
14 | 13 | anbi2d 460 |
. . . . . . 7
⊢ (𝑦 = [〈𝐶, 𝐷〉] ∼ →
(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ↔
([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼
))) |
15 | 14 | anbi1d 461 |
. . . . . 6
⊢ (𝑦 = [〈𝐶, 𝐷〉] ∼ →
((([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑) ↔ (([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))) |
16 | 15 | 4exbidv 1850 |
. . . . 5
⊢ (𝑦 = [〈𝐶, 𝐷〉] ∼ →
(∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))) |
17 | 12, 16 | opelopab2 4230 |
. . . 4
⊢
(([〈𝐴, 𝐵〉] ∼ ∈ 𝐻 ∧ [〈𝐶, 𝐷〉] ∼ ∈ 𝐻) → (〈[〈𝐴, 𝐵〉] ∼ , [〈𝐶, 𝐷〉] ∼ 〉 ∈
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [〈𝑧, 𝑤〉] ∼ ∧ 𝑦 = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))} ↔ ∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))) |
18 | 8, 17 | syl5bb 191 |
. . 3
⊢
(([〈𝐴, 𝐵〉] ∼ ∈ 𝐻 ∧ [〈𝐶, 𝐷〉] ∼ ∈ 𝐻) → ([〈𝐴, 𝐵〉] ∼ ≤ [〈𝐶, 𝐷〉] ∼ ↔ ∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))) |
19 | 3, 4, 18 | syl2an 287 |
. 2
⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([〈𝐴, 𝐵〉] ∼ ≤ [〈𝐶, 𝐷〉] ∼ ↔ ∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑))) |
20 | | opeq12 3743 |
. . . . . 6
⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → 〈𝑧, 𝑤〉 = 〈𝐴, 𝐵〉) |
21 | 20 | eceq1d 6516 |
. . . . 5
⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → [〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ) |
22 | | opeq12 3743 |
. . . . . 6
⊢ ((𝑣 = 𝐶 ∧ 𝑢 = 𝐷) → 〈𝑣, 𝑢〉 = 〈𝐶, 𝐷〉) |
23 | 22 | eceq1d 6516 |
. . . . 5
⊢ ((𝑣 = 𝐶 ∧ 𝑢 = 𝐷) → [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) |
24 | 21, 23 | anim12i 336 |
. . . 4
⊢ (((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) ∧ (𝑣 = 𝐶 ∧ 𝑢 = 𝐷)) → ([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ )) |
25 | | opelxpi 4618 |
. . . . . . . 8
⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) → 〈𝐴, 𝐵〉 ∈ (𝐺 × 𝐺)) |
26 | | opelxp 4616 |
. . . . . . . . 9
⊢
(〈𝑧, 𝑤〉 ∈ (𝐺 × 𝐺) ↔ (𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺)) |
27 | | brecop.2 |
. . . . . . . . . . 11
⊢ ∼ Er
(𝐺 × 𝐺) |
28 | 27 | a1i 9 |
. . . . . . . . . 10
⊢
([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ → ∼ Er
(𝐺 × 𝐺)) |
29 | | id 19 |
. . . . . . . . . 10
⊢
([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ → [〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ) |
30 | 28, 29 | ereldm 6523 |
. . . . . . . . 9
⊢
([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ → (〈𝑧, 𝑤〉 ∈ (𝐺 × 𝐺) ↔ 〈𝐴, 𝐵〉 ∈ (𝐺 × 𝐺))) |
31 | 26, 30 | bitr3id 193 |
. . . . . . . 8
⊢
([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ → ((𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺) ↔ 〈𝐴, 𝐵〉 ∈ (𝐺 × 𝐺))) |
32 | 25, 31 | syl5ibr 155 |
. . . . . . 7
⊢
([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ → ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) → (𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺))) |
33 | | opelxpi 4618 |
. . . . . . . 8
⊢ ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → 〈𝐶, 𝐷〉 ∈ (𝐺 × 𝐺)) |
34 | | opelxp 4616 |
. . . . . . . . 9
⊢
(〈𝑣, 𝑢〉 ∈ (𝐺 × 𝐺) ↔ (𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺)) |
35 | 27 | a1i 9 |
. . . . . . . . . 10
⊢
([〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ → ∼ Er
(𝐺 × 𝐺)) |
36 | | id 19 |
. . . . . . . . . 10
⊢
([〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ → [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) |
37 | 35, 36 | ereldm 6523 |
. . . . . . . . 9
⊢
([〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ → (〈𝑣, 𝑢〉 ∈ (𝐺 × 𝐺) ↔ 〈𝐶, 𝐷〉 ∈ (𝐺 × 𝐺))) |
38 | 34, 37 | bitr3id 193 |
. . . . . . . 8
⊢
([〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ → ((𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺) ↔ 〈𝐶, 𝐷〉 ∈ (𝐺 × 𝐺))) |
39 | 33, 38 | syl5ibr 155 |
. . . . . . 7
⊢
([〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ → ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → (𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺))) |
40 | 32, 39 | im2anan9 588 |
. . . . . 6
⊢
(([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ((𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺) ∧ (𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺)))) |
41 | | brecop.6 |
. . . . . . . . 9
⊢ ((((𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺) ∧ (𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺)) ∧ ((𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (𝜑 ↔ 𝜓))) |
42 | 41 | an4s 578 |
. . . . . . . 8
⊢ ((((𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺) ∧ (𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺)) ∧ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺))) → (([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (𝜑 ↔ 𝜓))) |
43 | 42 | ex 114 |
. . . . . . 7
⊢ (((𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺) ∧ (𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺)) → (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (𝜑 ↔ 𝜓)))) |
44 | 43 | com13 80 |
. . . . . 6
⊢
(([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (((𝑧 ∈ 𝐺 ∧ 𝑤 ∈ 𝐺) ∧ (𝑣 ∈ 𝐺 ∧ 𝑢 ∈ 𝐺)) → (𝜑 ↔ 𝜓)))) |
45 | 40, 44 | mpdd 41 |
. . . . 5
⊢
(([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (𝜑 ↔ 𝜓))) |
46 | 45 | pm5.74d 181 |
. . . 4
⊢
(([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) → ((((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → 𝜑) ↔ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → 𝜓))) |
47 | 24, 46 | cgsex4g 2749 |
. . 3
⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) ∧ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → 𝜑)) ↔ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → 𝜓))) |
48 | | eqcom 2159 |
. . . . . . 7
⊢
([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ↔ [〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ) |
49 | | eqcom 2159 |
. . . . . . 7
⊢
([〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ↔ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) |
50 | 48, 49 | anbi12i 456 |
. . . . . 6
⊢
(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ↔
([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ )) |
51 | 50 | a1i 9 |
. . . . 5
⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ↔
([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼
))) |
52 | | biimt 240 |
. . . . 5
⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (𝜑 ↔ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → 𝜑))) |
53 | 51, 52 | anbi12d 465 |
. . . 4
⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ((([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑) ↔ (([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) ∧ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → 𝜑)))) |
54 | 53 | 4exbidv 1850 |
. . 3
⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑) ↔ ∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝑧, 𝑤〉] ∼ = [〈𝐴, 𝐵〉] ∼ ∧ [〈𝑣, 𝑢〉] ∼ = [〈𝐶, 𝐷〉] ∼ ) ∧ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → 𝜑)))) |
55 | | biimt 240 |
. . 3
⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (𝜓 ↔ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → 𝜓))) |
56 | 47, 54, 55 | 3bitr4d 219 |
. 2
⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → (∃𝑧∃𝑤∃𝑣∃𝑢(([〈𝐴, 𝐵〉] ∼ = [〈𝑧, 𝑤〉] ∼ ∧ [〈𝐶, 𝐷〉] ∼ = [〈𝑣, 𝑢〉] ∼ ) ∧ 𝜑) ↔ 𝜓)) |
57 | 19, 56 | bitrd 187 |
1
⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([〈𝐴, 𝐵〉] ∼ ≤ [〈𝐶, 𝐷〉] ∼ ↔ 𝜓)) |