Theorem ceqsex8v 2900

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		19.42vv 1907	. . . . . . 7 ⊢ (∃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)) ∧ ∃t∃s((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ φ)))
2	1	2exbii 1583	. . . . . 6 ⊢ (∃v∃u∃t∃s(((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ ((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ φ)) ↔ ∃v∃u(((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ ∃t∃s((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ φ)))
3		19.42vv 1907	. . . . . 6 ⊢ (∃v∃u(((x = A ∧ y = B) ∧ (z = C ∧ w = D)) ∧ ∃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)))
4	2, 3	bitri 240	. . . . 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)))
5		3anass 938	. . . . . . . 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)))
6		df-3an 936	. . . . . . . . 9 ⊢ (((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ φ) ↔ (((v = E ∧ u = F) ∧ (t = G ∧ s = H)) ∧ φ))
7	6	anbi2i 675	. . . . . . . 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)))
8	5, 7	bitr4i 243	. . . . . . 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)))
9	8	2exbii 1583	. . . . . 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)))
10	9	2exbii 1583	. . . . 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)))
11		df-3an 936	. . . . 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)))
12	4, 10, 11	3bitr4i 268	. . . 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)))
13	12	2exbii 1583	. . 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)))
14	13	2exbii 1583	. 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)))
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 → (φ ↔ ψ))
20	19	3anbi3d 1258	. . . . 5 ⊢ (x = A → (((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ φ) ↔ ((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ ψ)))
21	20	4exbidv 1630	. . . 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)))
22		ceqsex8v.10	. . . . . 6 ⊢ (y = B → (ψ ↔ χ))
23	22	3anbi3d 1258	. . . . 5 ⊢ (y = B → (((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ ψ) ↔ ((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ χ)))
24	23	4exbidv 1630	. . . 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)))
25		ceqsex8v.11	. . . . . 6 ⊢ (z = C → (χ ↔ θ))
26	25	3anbi3d 1258	. . . . 5 ⊢ (z = C → (((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ χ) ↔ ((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ θ)))
27	26	4exbidv 1630	. . . 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)))
28		ceqsex8v.12	. . . . . 6 ⊢ (w = D → (θ ↔ τ))
29	28	3anbi3d 1258	. . . . 5 ⊢ (w = D → (((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ θ) ↔ ((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ τ)))
30	29	4exbidv 1630	. . . 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)))
31	15, 16, 17, 18, 21, 24, 27, 30	ceqsex4v 2898	. . 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))
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 → (τ ↔ η))
37		ceqsex8v.14	. . . 4 ⊢ (u = F → (η ↔ ζ))
38		ceqsex8v.15	. . . 4 ⊢ (t = G → (ζ ↔ σ))
39		ceqsex8v.16	. . . 4 ⊢ (s = H → (σ ↔ ρ))
40	32, 33, 34, 35, 36, 37, 38, 39	ceqsex4v 2898	. . 3 ⊢ (∃v∃u∃t∃s((v = E ∧ u = F) ∧ (t = G ∧ s = H) ∧ τ) ↔ ρ)
41	31, 40	bitri 240	. 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)
42	14, 41	bitri 240	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   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859

This theorem was proved from 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 theorem 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 2861

This theorem is referenced by: (None)