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Theorem ceqsex6v 2615
Description: Elimination of six existential quantifiers, using implicit substitution. (Contributed by NM, 21-Sep-2011.)
Hypotheses
Ref Expression
ceqsex6v.1 𝐴 ∈ V
ceqsex6v.2 𝐵 ∈ V
ceqsex6v.3 𝐶 ∈ V
ceqsex6v.4 𝐷 ∈ V
ceqsex6v.5 𝐸 ∈ V
ceqsex6v.6 𝐹 ∈ V
ceqsex6v.7 (𝑥 = 𝐴 → (𝜑𝜓))
ceqsex6v.8 (𝑦 = 𝐵 → (𝜓𝜒))
ceqsex6v.9 (𝑧 = 𝐶 → (𝜒𝜃))
ceqsex6v.10 (𝑤 = 𝐷 → (𝜃𝜏))
ceqsex6v.11 (𝑣 = 𝐸 → (𝜏𝜂))
ceqsex6v.12 (𝑢 = 𝐹 → (𝜂𝜁))
Assertion
Ref Expression
ceqsex6v (∃𝑥𝑦𝑧𝑤𝑣𝑢((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ (𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ 𝜁)
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,𝐸,𝑦,𝑧,𝑤,𝑣,𝑢   𝑥,𝐹,𝑦,𝑧,𝑤,𝑣,𝑢   𝜓,𝑥   𝜒,𝑦   𝜃,𝑧   𝜏,𝑤   𝜂,𝑣   𝜁,𝑢
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤,𝑣,𝑢)   𝜓(𝑦,𝑧,𝑤,𝑣,𝑢)   𝜒(𝑥,𝑧,𝑤,𝑣,𝑢)   𝜃(𝑥,𝑦,𝑤,𝑣,𝑢)   𝜏(𝑥,𝑦,𝑧,𝑣,𝑢)   𝜂(𝑥,𝑦,𝑧,𝑤,𝑢)   𝜁(𝑥,𝑦,𝑧,𝑤,𝑣)

Proof of Theorem ceqsex6v
StepHypRef Expression
1 3anass 900 . . . . 5 (((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ (𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ ((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑)))
213exbii 1514 . . . 4 (∃𝑤𝑣𝑢((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ (𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ ∃𝑤𝑣𝑢((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ ((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑)))
3 19.42vvv 1805 . . . 4 (∃𝑤𝑣𝑢((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ ((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑)) ↔ ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ ∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑)))
42, 3bitri 177 . . 3 (∃𝑤𝑣𝑢((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ (𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ ∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑)))
543exbii 1514 . 2 (∃𝑥𝑦𝑧𝑤𝑣𝑢((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ (𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ ∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ ∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑)))
6 ceqsex6v.1 . . . 4 𝐴 ∈ V
7 ceqsex6v.2 . . . 4 𝐵 ∈ V
8 ceqsex6v.3 . . . 4 𝐶 ∈ V
9 ceqsex6v.7 . . . . . 6 (𝑥 = 𝐴 → (𝜑𝜓))
109anbi2d 445 . . . . 5 (𝑥 = 𝐴 → (((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ ((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜓)))
11103exbidv 1765 . . . 4 (𝑥 = 𝐴 → (∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ ∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜓)))
12 ceqsex6v.8 . . . . . 6 (𝑦 = 𝐵 → (𝜓𝜒))
1312anbi2d 445 . . . . 5 (𝑦 = 𝐵 → (((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜓) ↔ ((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜒)))
14133exbidv 1765 . . . 4 (𝑦 = 𝐵 → (∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜓) ↔ ∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜒)))
15 ceqsex6v.9 . . . . . 6 (𝑧 = 𝐶 → (𝜒𝜃))
1615anbi2d 445 . . . . 5 (𝑧 = 𝐶 → (((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜒) ↔ ((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜃)))
17163exbidv 1765 . . . 4 (𝑧 = 𝐶 → (∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜒) ↔ ∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜃)))
186, 7, 8, 11, 14, 17ceqsex3v 2613 . . 3 (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ ∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑)) ↔ ∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜃))
19 ceqsex6v.4 . . . 4 𝐷 ∈ V
20 ceqsex6v.5 . . . 4 𝐸 ∈ V
21 ceqsex6v.6 . . . 4 𝐹 ∈ V
22 ceqsex6v.10 . . . 4 (𝑤 = 𝐷 → (𝜃𝜏))
23 ceqsex6v.11 . . . 4 (𝑣 = 𝐸 → (𝜏𝜂))
24 ceqsex6v.12 . . . 4 (𝑢 = 𝐹 → (𝜂𝜁))
2519, 20, 21, 22, 23, 24ceqsex3v 2613 . . 3 (∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜃) ↔ 𝜁)
2618, 25bitri 177 . 2 (∃𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ ∃𝑤𝑣𝑢((𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑)) ↔ 𝜁)
275, 26bitri 177 1 (∃𝑥𝑦𝑧𝑤𝑣𝑢((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ∧ (𝑤 = 𝐷𝑣 = 𝐸𝑢 = 𝐹) ∧ 𝜑) ↔ 𝜁)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by: (None)
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