Theorem ceqsex8v 3509

Description: Elimination of eight existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Hypotheses

Ref	Expression
ceqsex8v.1	⊢ 𝐴 ∈ V
ceqsex8v.2	⊢ 𝐵 ∈ V
ceqsex8v.3	⊢ 𝐶 ∈ V
ceqsex8v.4	⊢ 𝐷 ∈ V
ceqsex8v.5	⊢ 𝐸 ∈ V
ceqsex8v.6	⊢ 𝐹 ∈ V
ceqsex8v.7	⊢ 𝐺 ∈ V
ceqsex8v.8	⊢ 𝐻 ∈ V
ceqsex8v.9	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
ceqsex8v.10	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
ceqsex8v.11	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
ceqsex8v.12	⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏))
ceqsex8v.13	⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂))
ceqsex8v.14	⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁))
ceqsex8v.15	⊢ (𝑡 = 𝐺 → (𝜁 ↔ 𝜎))
ceqsex8v.16	⊢ (𝑠 = 𝐻 → (𝜎 ↔ 𝜌))

Assertion

Ref	Expression
ceqsex8v	⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌)

Distinct variable groups:   𝑥,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠,𝐴   𝑥,𝐵,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐶,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐷,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐸,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐹,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐺,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝑥,𝐻,𝑦,𝑧,𝑤,𝑣,𝑢,𝑡,𝑠   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧   𝜏,𝑤   𝜂,𝑣   𝜁,𝑢   𝜎,𝑡   𝜌,𝑠

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, 𝑠)   𝜓(𝑦, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, 𝑠)   𝜒(𝑥, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡, 𝑠)   𝜃(𝑥, 𝑦, 𝑤, 𝑣, 𝑢, 𝑡, 𝑠)   𝜏(𝑥, 𝑦, 𝑧, 𝑣, 𝑢, 𝑡, 𝑠)   𝜂(𝑥, 𝑦, 𝑧, 𝑤, 𝑢, 𝑡, 𝑠)   𝜁(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑡, 𝑠)   𝜎(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, 𝑠)   𝜌(𝑥, 𝑦, 𝑧, 𝑤, 𝑣, 𝑢, 𝑡)

Proof of Theorem ceqsex8v

Step	Hyp	Ref	Expression
1		19.42vv 1986	. . . . . . 7 ⊢ (∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
2	1	2exbii 1878	. . . . . 6 ⊢ (∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)) ↔ ∃𝑣∃𝑢(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
3		19.42vv 1986	. . . . . 6 ⊢ (∃𝑣∃𝑢(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
4	2, 3	bitri 278	. . . . 5 ⊢ (∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
5		3anass 1110	. . . . . . . 8 ⊢ ((((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑)))
6		df-3an 1104	. . . . . . . . 9 ⊢ (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑) ↔ (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑))
7	6	anbi2i 634	. . . . . . . 8 ⊢ ((((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑)))
8	5, 7	bitr4i 281	. . . . . . 7 ⊢ ((((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
9	8	2exbii 1878	. . . . . 6 ⊢ (∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
10	9	2exbii 1878	. . . . 5 ⊢ (∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
11		df-3an 1104	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
12	4, 10, 11	3bitr4i 306	. . . 4 ⊢ (∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
13	12	2exbii 1878	. . 3 ⊢ (∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
14	13	2exbii 1878	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)))
15		ceqsex8v.1	. . 3 ⊢ 𝐴 ∈ V
16		ceqsex8v.2	. . 3 ⊢ 𝐵 ∈ V
17		ceqsex8v.3	. . 3 ⊢ 𝐶 ∈ V
18		ceqsex8v.4	. . 3 ⊢ 𝐷 ∈ V
19		ceqsex8v.9	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
20	19	3anbi3d 1469	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑) ↔ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜓)))
21	20	4exbidv 1955	. . 3 ⊢ (𝑥 = 𝐴 → (∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑) ↔ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜓)))
22		ceqsex8v.10	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
23	22	3anbi3d 1469	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜓) ↔ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜒)))
24	23	4exbidv 1955	. . 3 ⊢ (𝑦 = 𝐵 → (∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜓) ↔ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜒)))
25		ceqsex8v.11	. . . . 5 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
26	25	3anbi3d 1469	. . . 4 ⊢ (𝑧 = 𝐶 → (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜒) ↔ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜃)))
27	26	4exbidv 1955	. . 3 ⊢ (𝑧 = 𝐶 → (∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜒) ↔ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜃)))
28		ceqsex8v.12	. . . . 5 ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏))
29	28	3anbi3d 1469	. . . 4 ⊢ (𝑤 = 𝐷 → (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜃) ↔ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜏)))
30	29	4exbidv 1955	. . 3 ⊢ (𝑤 = 𝐷 → (∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜃) ↔ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜏)))
31	15, 16, 17, 18, 21, 24, 27, 30	ceqsex4v 3507	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷) ∧ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑)) ↔ ∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜏))
32		ceqsex8v.5	. . 3 ⊢ 𝐸 ∈ V
33		ceqsex8v.6	. . 3 ⊢ 𝐹 ∈ V
34		ceqsex8v.7	. . 3 ⊢ 𝐺 ∈ V
35		ceqsex8v.8	. . 3 ⊢ 𝐻 ∈ V
36		ceqsex8v.13	. . 3 ⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂))
37		ceqsex8v.14	. . 3 ⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁))
38		ceqsex8v.15	. . 3 ⊢ (𝑡 = 𝐺 → (𝜁 ↔ 𝜎))
39		ceqsex8v.16	. . 3 ⊢ (𝑠 = 𝐻 → (𝜎 ↔ 𝜌))
40	32, 33, 34, 35, 36, 37, 38, 39	ceqsex4v 3507	. 2 ⊢ (∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜏) ↔ 𝜌)
41	14, 31, 40	3bitri 300	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢∃𝑡∃𝑠(((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ∧ (𝑧 = 𝐶 ∧ 𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102   = wceq 1569  ∃wex 1808   ∈ wcel 2142  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-ex 1809  df-clel 2837

This theorem is used by: (None)