Theorem ceqsex6v 2900

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 938	. . . . 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 1584	. . . 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 1908	. . . 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 240	. . 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 1584	. 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 684	. . . . 5 ⊢ (x = A → (((w = D ∧ v = E ∧ u = F) ∧ φ) ↔ ((w = D ∧ v = E ∧ u = F) ∧ ψ)))
11	10	3exbidv 1629	. . . 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 684	. . . . 5 ⊢ (y = B → (((w = D ∧ v = E ∧ u = F) ∧ ψ) ↔ ((w = D ∧ v = E ∧ u = F) ∧ χ)))
14	13	3exbidv 1629	. . . 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 684	. . . . 5 ⊢ (z = C → (((w = D ∧ v = E ∧ u = F) ∧ χ) ↔ ((w = D ∧ v = E ∧ u = F) ∧ θ)))
17	16	3exbidv 1629	. . . 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 2898	. . 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 2898	. . 3 ⊢ (∃w∃v∃u((w = D ∧ v = E ∧ u = F) ∧ θ) ↔ ζ)
26	18, 25	bitri 240	. 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 240	1 ⊢ (∃x∃y∃z∃w∃v∃u((x = A ∧ y = B ∧ z = C) ∧ (w = D ∧ v = E ∧ u = F) ∧ φ) ↔ ζ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is used by: (None)