Theorem ceqsex6v 3462

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 → (𝜑 ↔ 𝜓))
ceqsex6v.8	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
ceqsex6v.9	⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
ceqsex6v.10	⊢ (𝑤 = 𝐷 → (𝜃 ↔ 𝜏))
ceqsex6v.11	⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂))
ceqsex6v.12	⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁))

Assertion

Ref	Expression
ceqsex6v	⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ (𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 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

Step	Hyp	Ref	Expression
1		3anass 1097	. . . . 5 ⊢ (((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ (𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ ((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑)))
2	1	3exbii 1857	. . . 4 ⊢ (∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ (𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ ∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ ((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑)))
3		19.42vvv 1968	. . . 4 ⊢ (∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ ((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑)) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ ∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑)))
4	2, 3	bitri 278	. . 3 ⊢ (∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ (𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ ∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑)))
5	4	3exbii 1857	. 2 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ (𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ ∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ ∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 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 → (𝜑 ↔ 𝜓))
10	9	anbi2d 632	. . . . 5 ⊢ (𝑥 = 𝐴 → (((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ ((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜓)))
11	10	3exbidv 1933	. . . 4 ⊢ (𝑥 = 𝐴 → (∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ ∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜓)))
12		ceqsex6v.8	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
13	12	anbi2d 632	. . . . 5 ⊢ (𝑦 = 𝐵 → (((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜓) ↔ ((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜒)))
14	13	3exbidv 1933	. . . 4 ⊢ (𝑦 = 𝐵 → (∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜓) ↔ ∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜒)))
15		ceqsex6v.9	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝜒 ↔ 𝜃))
16	15	anbi2d 632	. . . . 5 ⊢ (𝑧 = 𝐶 → (((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜒) ↔ ((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜃)))
17	16	3exbidv 1933	. . . 4 ⊢ (𝑧 = 𝐶 → (∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜒) ↔ ∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜃)))
18	6, 7, 8, 11, 14, 17	ceqsex3v 3460	. . 3 ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ ∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑)) ↔ ∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 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 → (𝜃 ↔ 𝜏))
23		ceqsex6v.11	. . . 4 ⊢ (𝑣 = 𝐸 → (𝜏 ↔ 𝜂))
24		ceqsex6v.12	. . . 4 ⊢ (𝑢 = 𝐹 → (𝜂 ↔ 𝜁))
25	19, 20, 21, 22, 23, 24	ceqsex3v 3460	. . 3 ⊢ (∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜃) ↔ 𝜁)
26	18, 25	bitri 278	. 2 ⊢ (∃𝑥∃𝑦∃𝑧((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ ∃𝑤∃𝑣∃𝑢((𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑)) ↔ 𝜁)
27	5, 26	bitri 278	1 ⊢ (∃𝑥∃𝑦∃𝑧∃𝑤∃𝑣∃𝑢((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ (𝑤 = 𝐷 ∧ 𝑣 = 𝐸 ∧ 𝑢 = 𝐹) ∧ 𝜑) ↔ 𝜁)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543  ∃wex 1787   ∈ wcel 2110  Vcvv 3408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112

This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1091  df-ex 1788  df-clel 2816

This theorem is referenced by: (None)