Theorem ceqsex4v 2899

Description: Elimination of four existential quantifiers, using implicit substitution. (Contributed by NM, 23-Sep-2011.)

Hypotheses

Ref	Expression
ceqsex4v.1	⊢ A ∈ V
ceqsex4v.2	⊢ B ∈ V
ceqsex4v.3	⊢ C ∈ V
ceqsex4v.4	⊢ D ∈ V
ceqsex4v.7	⊢ (x = A → (φ ↔ ψ))
ceqsex4v.8	⊢ (y = B → (ψ ↔ χ))
ceqsex4v.9	⊢ (z = C → (χ ↔ θ))
ceqsex4v.10	⊢ (w = D → (θ ↔ τ))

Assertion

Ref	Expression
ceqsex4v	⊢ (∃x∃y∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D) ∧ φ) ↔ τ)

Distinct variable groups:   x,y,z,w,A   x,B,y,z,w   x,C,y,z,w   x,D,y,z,w   ψ,x   χ,y   θ,z   τ,w

Allowed substitution hints:   φ(x,y,z,w)   ψ(y,z,w)   χ(x,z,w)   θ(x,y,w)   τ(x,y,z)

Proof of Theorem ceqsex4v

Step	Hyp	Ref	Expression
1		19.42vv 1907	. . . 4 ⊢ (∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D ∧ φ)) ↔ ((x = A ∧ y = B) ∧ ∃z∃w(z = C ∧ w = D ∧ φ)))
2		3anass 938	. . . . . 6 ⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D) ∧ φ) ↔ ((x = A ∧ y = B) ∧ ((z = C ∧ w = D) ∧ φ)))
3		df-3an 936	. . . . . . 7 ⊢ ((z = C ∧ w = D ∧ φ) ↔ ((z = C ∧ w = D) ∧ φ))
4	3	anbi2i 675	. . . . . 6 ⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D ∧ φ)) ↔ ((x = A ∧ y = B) ∧ ((z = C ∧ w = D) ∧ φ)))
5	2, 4	bitr4i 243	. . . . 5 ⊢ (((x = A ∧ y = B) ∧ (z = C ∧ w = D) ∧ φ) ↔ ((x = A ∧ y = B) ∧ (z = C ∧ w = D ∧ φ)))
6	5	2exbii 1583	. . . 4 ⊢ (∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D) ∧ φ) ↔ ∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D ∧ φ)))
7		df-3an 936	. . . 4 ⊢ ((x = A ∧ y = B ∧ ∃z∃w(z = C ∧ w = D ∧ φ)) ↔ ((x = A ∧ y = B) ∧ ∃z∃w(z = C ∧ w = D ∧ φ)))
8	1, 6, 7	3bitr4i 268	. . 3 ⊢ (∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D) ∧ φ) ↔ (x = A ∧ y = B ∧ ∃z∃w(z = C ∧ w = D ∧ φ)))
9	8	2exbii 1583	. 2 ⊢ (∃x∃y∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D) ∧ φ) ↔ ∃x∃y(x = A ∧ y = B ∧ ∃z∃w(z = C ∧ w = D ∧ φ)))
10		ceqsex4v.1	. . 3 ⊢ A ∈ V
11		ceqsex4v.2	. . 3 ⊢ B ∈ V
12		ceqsex4v.7	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
13	12	3anbi3d 1258	. . . 4 ⊢ (x = A → ((z = C ∧ w = D ∧ φ) ↔ (z = C ∧ w = D ∧ ψ)))
14	13	2exbidv 1628	. . 3 ⊢ (x = A → (∃z∃w(z = C ∧ w = D ∧ φ) ↔ ∃z∃w(z = C ∧ w = D ∧ ψ)))
15		ceqsex4v.8	. . . . 5 ⊢ (y = B → (ψ ↔ χ))
16	15	3anbi3d 1258	. . . 4 ⊢ (y = B → ((z = C ∧ w = D ∧ ψ) ↔ (z = C ∧ w = D ∧ χ)))
17	16	2exbidv 1628	. . 3 ⊢ (y = B → (∃z∃w(z = C ∧ w = D ∧ ψ) ↔ ∃z∃w(z = C ∧ w = D ∧ χ)))
18	10, 11, 14, 17	ceqsex2v 2897	. 2 ⊢ (∃x∃y(x = A ∧ y = B ∧ ∃z∃w(z = C ∧ w = D ∧ φ)) ↔ ∃z∃w(z = C ∧ w = D ∧ χ))
19		ceqsex4v.3	. . 3 ⊢ C ∈ V
20		ceqsex4v.4	. . 3 ⊢ D ∈ V
21		ceqsex4v.9	. . 3 ⊢ (z = C → (χ ↔ θ))
22		ceqsex4v.10	. . 3 ⊢ (w = D → (θ ↔ τ))
23	19, 20, 21, 22	ceqsex2v 2897	. 2 ⊢ (∃z∃w(z = C ∧ w = D ∧ χ) ↔ τ)
24	9, 18, 23	3bitri 262	1 ⊢ (∃x∃y∃z∃w((x = A ∧ y = B) ∧ (z = C ∧ w = D) ∧ φ) ↔ τ)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860

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 2862

This theorem is referenced by:  ceqsex8v  2901