Theorem ceqsex8v 2775

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		19.42vvvv 1906	. . . . 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)))
2		3anass 977	. . . . . . . 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)))
3		df-3an 975	. . . . . . . . 9 ⊢ (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑) ↔ (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻)) ∧ 𝜑))
4	3	anbi2i 454	. . . . . . . 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)))
5	2, 4	bitr4i 186	. . . . . . 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)))
6	5	2exbii 1599	. . . . . 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)))
7	6	2exbii 1599	. . . . 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)))
8		df-3an 975	. . . . 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)))
9	1, 7, 8	3bitr4i 211	. . . 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)))
10	9	2exbii 1599	. . 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)))
11	10	2exbii 1599	. 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)))
12		ceqsex8v.1	. . . 4 ⊢ 𝐴 ∈ V
13		ceqsex8v.2	. . . 4 ⊢ 𝐵 ∈ V
14		ceqsex8v.3	. . . 4 ⊢ 𝐶 ∈ V
15		ceqsex8v.4	. . . 4 ⊢ 𝐷 ∈ V
16		ceqsex8v.9	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
17	16	3anbi3d 1313	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜑) ↔ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜓)))
18	17	4exbidv 1863	. . . 4 ⊢ (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)))
19		ceqsex8v.10	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
20	19	3anbi3d 1313	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜓) ↔ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜒)))
21	20	4exbidv 1863	. . . 4 ⊢ (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)))
22		ceqsex8v.11	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
23	22	3anbi3d 1313	. . . . 5 ⊢ (𝑧 = 𝐶 → (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜒) ↔ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜃)))
24	23	4exbidv 1863	. . . 4 ⊢ (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)))
25		ceqsex8v.12	. . . . . 6 ⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏))
26	25	3anbi3d 1313	. . . . 5 ⊢ (𝑤 = 𝐷 → (((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜃) ↔ ((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜏)))
27	26	4exbidv 1863	. . . 4 ⊢ (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)))
28	12, 13, 14, 15, 18, 21, 24, 27	ceqsex4v 2773	. . 3 ⊢ (∃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))
29		ceqsex8v.5	. . . 4 ⊢ 𝐸 ∈ V
30		ceqsex8v.6	. . . 4 ⊢ 𝐹 ∈ V
31		ceqsex8v.7	. . . 4 ⊢ 𝐺 ∈ V
32		ceqsex8v.8	. . . 4 ⊢ 𝐻 ∈ V
33		ceqsex8v.13	. . . 4 ⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂))
34		ceqsex8v.14	. . . 4 ⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁))
35		ceqsex8v.15	. . . 4 ⊢ (𝑡 = 𝐺 → (𝜁 ↔ 𝜎))
36		ceqsex8v.16	. . . 4 ⊢ (𝑠 = 𝐻 → (𝜎 ↔ 𝜌))
37	29, 30, 31, 32, 33, 34, 35, 36	ceqsex4v 2773	. . 3 ⊢ (∃𝑣∃𝑢∃𝑡∃𝑠((𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ (𝑡 = 𝐺 ∧ 𝑠 = 𝐻) ∧ 𝜏) ↔ 𝜌)
38	28, 37	bitri 183	. 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)) ↔ 𝜌)
39	11, 38	bitri 183	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)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   ∧ w3a 973   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730

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-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by: (None)