Theorem ceqsex3v 2897

Description: Elimination of three existential quantifiers, using implicit substitution. (Contributed by NM, 16-Aug-2011.)

Hypotheses

Ref	Expression
ceqsex3v.1	⊢ A ∈ V
ceqsex3v.2	⊢ B ∈ V
ceqsex3v.3	⊢ C ∈ V
ceqsex3v.4	⊢ (x = A → (φ ↔ ψ))
ceqsex3v.5	⊢ (y = B → (ψ ↔ χ))
ceqsex3v.6	⊢ (z = C → (χ ↔ θ))

Assertion

Ref	Expression
ceqsex3v	⊢ (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ θ)

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

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

Proof of Theorem ceqsex3v

Step	Hyp	Ref	Expression
1		anass 630	. . . . . 6 ⊢ (((x = A ∧ (y = B ∧ z = C)) ∧ φ) ↔ (x = A ∧ ((y = B ∧ z = C) ∧ φ)))
2		3anass 938	. . . . . . 7 ⊢ ((x = A ∧ y = B ∧ z = C) ↔ (x = A ∧ (y = B ∧ z = C)))
3	2	anbi1i 676	. . . . . 6 ⊢ (((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ ((x = A ∧ (y = B ∧ z = C)) ∧ φ))
4		df-3an 936	. . . . . . 7 ⊢ ((y = B ∧ z = C ∧ φ) ↔ ((y = B ∧ z = C) ∧ φ))
5	4	anbi2i 675	. . . . . 6 ⊢ ((x = A ∧ (y = B ∧ z = C ∧ φ)) ↔ (x = A ∧ ((y = B ∧ z = C) ∧ φ)))
6	1, 3, 5	3bitr4i 268	. . . . 5 ⊢ (((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ (x = A ∧ (y = B ∧ z = C ∧ φ)))
7	6	2exbii 1583	. . . 4 ⊢ (∃y∃z((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ ∃y∃z(x = A ∧ (y = B ∧ z = C ∧ φ)))
8		19.42vv 1907	. . . 4 ⊢ (∃y∃z(x = A ∧ (y = B ∧ z = C ∧ φ)) ↔ (x = A ∧ ∃y∃z(y = B ∧ z = C ∧ φ)))
9	7, 8	bitri 240	. . 3 ⊢ (∃y∃z((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ (x = A ∧ ∃y∃z(y = B ∧ z = C ∧ φ)))
10	9	exbii 1582	. 2 ⊢ (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ ∃x(x = A ∧ ∃y∃z(y = B ∧ z = C ∧ φ)))
11		ceqsex3v.1	. . . 4 ⊢ A ∈ V
12		ceqsex3v.4	. . . . . 6 ⊢ (x = A → (φ ↔ ψ))
13	12	3anbi3d 1258	. . . . 5 ⊢ (x = A → ((y = B ∧ z = C ∧ φ) ↔ (y = B ∧ z = C ∧ ψ)))
14	13	2exbidv 1628	. . . 4 ⊢ (x = A → (∃y∃z(y = B ∧ z = C ∧ φ) ↔ ∃y∃z(y = B ∧ z = C ∧ ψ)))
15	11, 14	ceqsexv 2894	. . 3 ⊢ (∃x(x = A ∧ ∃y∃z(y = B ∧ z = C ∧ φ)) ↔ ∃y∃z(y = B ∧ z = C ∧ ψ))
16		ceqsex3v.2	. . . 4 ⊢ B ∈ V
17		ceqsex3v.3	. . . 4 ⊢ C ∈ V
18		ceqsex3v.5	. . . 4 ⊢ (y = B → (ψ ↔ χ))
19		ceqsex3v.6	. . . 4 ⊢ (z = C → (χ ↔ θ))
20	16, 17, 18, 19	ceqsex2v 2896	. . 3 ⊢ (∃y∃z(y = B ∧ z = C ∧ ψ) ↔ θ)
21	15, 20	bitri 240	. 2 ⊢ (∃x(x = A ∧ ∃y∃z(y = B ∧ z = C ∧ φ)) ↔ θ)
22	10, 21	bitri 240	1 ⊢ (∃x∃y∃z((x = A ∧ y = B ∧ z = C) ∧ φ) ↔ θ)

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:  ceqsex6v  2899