Theorem brecop 6591

Description: Binary relation on a quotient set. Lemma for real number construction. (Contributed by NM, 29-Jan-1996.)

Hypotheses

Ref	Expression
brecop.1	⊢ ∼ ∈ V
brecop.2	⊢ ∼ Er (𝐺 × 𝐺)
brecop.4	⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ )
brecop.5	⊢ ≤ = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ 𝐻 ∧ 𝑦 ∈ 𝐻) ∧ ∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = [⟨𝑧, 𝑤⟩] ∼ ∧ 𝑦 = [⟨𝑣, 𝑢⟩] ∼ ) ∧ 𝜑))}
brecop.6	⊢ ((((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)))

Assertion

Ref	Expression
brecop	⊢ (((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) ∧ (𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺)) → ([⟨𝐴, 𝐵⟩] ∼ ≤ [⟨𝐶, 𝐷⟩] ∼ ↔ 𝜓))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐷,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥, ∼ ,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐻,𝑦   𝑧,𝐺,𝑤,𝑣,𝑢   𝜑,𝑥,𝑦   𝜓,𝑧,𝑤,𝑣,𝑢

Allowed substitution hints:   𝜑(𝑧,𝑤,𝑣,𝑢)   𝜓(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑧,𝑤,𝑣,𝑢)   ≤ (𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)

Proof of Theorem brecop

Step	Hyp	Ref	Expression
1		brecop.1	. . . 4 ⊢ ∼ ∈ V
2		brecop.4	. . . 4 ⊢ 𝐻 = ((𝐺 × 𝐺) / ∼ )
3	1, 2	ecopqsi 6556	. . 3 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) → [⟨𝐴, 𝐵⟩] ∼ ∈ 𝐻)
4	1, 2	ecopqsi 6556	. . 3 ⊢ ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → [⟨𝐶, 𝐷⟩] ∼ ∈ 𝐻)
5		df-br 3983	. . . . 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 2233	. . . . 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 2172	. . . . . . . 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 1858	. . . . 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 2172	. . . . . . . 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 1858	. . . . 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 4248	. . . 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 3760	. . . . . 6 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → ⟨𝑧, 𝑤⟩ = ⟨𝐴, 𝐵⟩)
21	20	eceq1d 6537	. . . . 5 ⊢ ((𝑧 = 𝐴 ∧ 𝑤 = 𝐵) → [⟨𝑧, 𝑤⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ )
22		opeq12 3760	. . . . . 6 ⊢ ((𝑣 = 𝐶 ∧ 𝑢 = 𝐷) → ⟨𝑣, 𝑢⟩ = ⟨𝐶, 𝐷⟩)
23	22	eceq1d 6537	. . . . 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 4636	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝐺 ∧ 𝐵 ∈ 𝐺) → ⟨𝐴, 𝐵⟩ ∈ (𝐺 × 𝐺))
26		opelxp 4634	. . . . . . . . 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 6544	. . . . . . . . 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 4636	. . . . . . . 8 ⊢ ((𝐶 ∈ 𝐺 ∧ 𝐷 ∈ 𝐺) → ⟨𝐶, 𝐷⟩ ∈ (𝐺 × 𝐺))
34		opelxp 4634	. . . . . . . . 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 6544	. . . . . . . . 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 2763	. . 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 2167	. . . . . . 7 ⊢ ([⟨𝐴, 𝐵⟩] ∼ = [⟨𝑧, 𝑤⟩] ∼ ↔ [⟨𝑧, 𝑤⟩] ∼ = [⟨𝐴, 𝐵⟩] ∼ )
49		eqcom 2167	. . . . . . 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 1858	. . 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))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726  ⟨cop 3579   class class class wbr 3982  {copab 4042   × cxp 4602   Er wer 6498  [cec 6499   / cqs 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-cnv 4612  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-er 6501  df-ec 6503  df-qs 6507

This theorem is referenced by:  ordpipqqs  7315  ltsrprg  7688