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Theorem ceqsex8v 3239
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 → (𝜑𝜓))
ceqsex8v.10 (𝑦 = 𝐵 → (𝜓𝜒))
ceqsex8v.11 (𝑧 = 𝐶 → (𝜒𝜃))
ceqsex8v.12 (𝑤 = 𝐷 → (𝜃𝜏))
ceqsex8v.13 (𝑣 = 𝐸 → (𝜏𝜂))
ceqsex8v.14 (𝑢 = 𝐹 → (𝜂𝜁))
ceqsex8v.15 (𝑡 = 𝐺 → (𝜁𝜎))
ceqsex8v.16 (𝑠 = 𝐻 → (𝜎𝜌))
Assertion
Ref Expression
ceqsex8v (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 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
StepHypRef Expression
1 19.42vv 1917 . . . . . . 7 (∃𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ∃𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
212exbii 1772 . . . . . 6 (∃𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ ∃𝑣𝑢(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ∃𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
3 19.42vv 1917 . . . . . 6 (∃𝑣𝑢(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ∃𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
42, 3bitri 264 . . . . 5 (∃𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
5 3anass 1040 . . . . . . . 8 ((((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑)))
6 df-3an 1038 . . . . . . . . 9 (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑) ↔ (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑))
76anbi2i 729 . . . . . . . 8 ((((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑)))
85, 7bitr4i 267 . . . . . . 7 ((((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
982exbii 1772 . . . . . 6 (∃𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
1092exbii 1772 . . . . 5 (∃𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
11 df-3an 1038 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ (((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
124, 10, 113bitr4i 292 . . . 4 (∃𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
13122exbii 1772 . . 3 (∃𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
14132exbii 1772 . 2 (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ ∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)))
15 ceqsex8v.1 . . . 4 𝐴 ∈ V
16 ceqsex8v.2 . . . 4 𝐵 ∈ V
17 ceqsex8v.3 . . . 4 𝐶 ∈ V
18 ceqsex8v.4 . . . 4 𝐷 ∈ V
19 ceqsex8v.9 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
20193anbi3d 1402 . . . . 5 (𝑥 = 𝐴 → (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑) ↔ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜓)))
21204exbidv 1851 . . . 4 (𝑥 = 𝐴 → (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜓)))
22 ceqsex8v.10 . . . . . 6 (𝑦 = 𝐵 → (𝜓𝜒))
23223anbi3d 1402 . . . . 5 (𝑦 = 𝐵 → (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜓) ↔ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜒)))
24234exbidv 1851 . . . 4 (𝑦 = 𝐵 → (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜓) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜒)))
25 ceqsex8v.11 . . . . . 6 (𝑧 = 𝐶 → (𝜒𝜃))
26253anbi3d 1402 . . . . 5 (𝑧 = 𝐶 → (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜒) ↔ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜃)))
27264exbidv 1851 . . . 4 (𝑧 = 𝐶 → (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜒) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜃)))
28 ceqsex8v.12 . . . . . 6 (𝑤 = 𝐷 → (𝜃𝜏))
29283anbi3d 1402 . . . . 5 (𝑤 = 𝐷 → (((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜃) ↔ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜏)))
30294exbidv 1851 . . . 4 (𝑤 = 𝐷 → (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜃) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜏)))
3115, 16, 17, 18, 21, 24, 27, 30ceqsex4v 3237 . . 3 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜏))
32 ceqsex8v.5 . . . 4 𝐸 ∈ V
33 ceqsex8v.6 . . . 4 𝐹 ∈ V
34 ceqsex8v.7 . . . 4 𝐺 ∈ V
35 ceqsex8v.8 . . . 4 𝐻 ∈ V
36 ceqsex8v.13 . . . 4 (𝑣 = 𝐸 → (𝜏𝜂))
37 ceqsex8v.14 . . . 4 (𝑢 = 𝐹 → (𝜂𝜁))
38 ceqsex8v.15 . . . 4 (𝑡 = 𝐺 → (𝜁𝜎))
39 ceqsex8v.16 . . . 4 (𝑠 = 𝐻 → (𝜎𝜌))
4032, 33, 34, 35, 36, 37, 38, 39ceqsex4v 3237 . . 3 (∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜏) ↔ 𝜌)
4131, 40bitri 264 . 2 (∃𝑥𝑦𝑧𝑤((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷) ∧ ∃𝑣𝑢𝑡𝑠((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻) ∧ 𝜑)) ↔ 𝜌)
4214, 41bitri 264 1 (∃𝑥𝑦𝑧𝑤𝑣𝑢𝑡𝑠(((𝑥 = 𝐴𝑦 = 𝐵) ∧ (𝑧 = 𝐶𝑤 = 𝐷)) ∧ ((𝑣 = 𝐸𝑢 = 𝐹) ∧ (𝑡 = 𝐺𝑠 = 𝐻)) ∧ 𝜑) ↔ 𝜌)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  Vcvv 3190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3192
This theorem is referenced by: (None)
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