Theorem vtocl3 2911

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
vtocl3.1	⊢ A ∈ V
vtocl3.2	⊢ B ∈ V
vtocl3.3	⊢ C ∈ V
vtocl3.4	⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))
vtocl3.5	⊢ φ

Assertion

Ref	Expression
vtocl3	⊢ ψ

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

Allowed substitution hints:   φ(x,y,z)

Proof of Theorem vtocl3

Step	Hyp	Ref	Expression
1		vtocl3.1	. . . . . . 7 ⊢ A ∈ V
2	1	isseti 2865	. . . . . 6 ⊢ ∃x x = A
3		vtocl3.2	. . . . . . 7 ⊢ B ∈ V
4	3	isseti 2865	. . . . . 6 ⊢ ∃y y = B
5		vtocl3.3	. . . . . . 7 ⊢ C ∈ V
6	5	isseti 2865	. . . . . 6 ⊢ ∃z z = C
7		eeeanv 1914	. . . . . . 7 ⊢ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) ↔ (∃x x = A ∧ ∃y y = B ∧ ∃z z = C))
8		vtocl3.4	. . . . . . . . . 10 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ ↔ ψ))
9	8	biimpd 198	. . . . . . . . 9 ⊢ ((x = A ∧ y = B ∧ z = C) → (φ → ψ))
10	9	eximi 1576	. . . . . . . 8 ⊢ (∃z(x = A ∧ y = B ∧ z = C) → ∃z(φ → ψ))
11	10	2eximi 1577	. . . . . . 7 ⊢ (∃x∃y∃z(x = A ∧ y = B ∧ z = C) → ∃x∃y∃z(φ → ψ))
12	7, 11	sylbir 204	. . . . . 6 ⊢ ((∃x x = A ∧ ∃y y = B ∧ ∃z z = C) → ∃x∃y∃z(φ → ψ))
13	2, 4, 6, 12	mp3an 1277	. . . . 5 ⊢ ∃x∃y∃z(φ → ψ)
14		19.36v 1896	. . . . . 6 ⊢ (∃z(φ → ψ) ↔ (∀zφ → ψ))
15	14	2exbii 1583	. . . . 5 ⊢ (∃x∃y∃z(φ → ψ) ↔ ∃x∃y(∀zφ → ψ))
16	13, 15	mpbi 199	. . . 4 ⊢ ∃x∃y(∀zφ → ψ)
17		19.36v 1896	. . . . 5 ⊢ (∃y(∀zφ → ψ) ↔ (∀y∀zφ → ψ))
18	17	exbii 1582	. . . 4 ⊢ (∃x∃y(∀zφ → ψ) ↔ ∃x(∀y∀zφ → ψ))
19	16, 18	mpbi 199	. . 3 ⊢ ∃x(∀y∀zφ → ψ)
20	19	19.36aiv 1897	. 2 ⊢ (∀x∀y∀zφ → ψ)
21		vtocl3.5	. . 3 ⊢ φ
22	21	gen2 1547	. 2 ⊢ ∀y∀zφ
23	20, 22	mpg 1548	1 ⊢ ψ

Syntax hints:   → wi 4   ↔ wb 176   ∧ w3a 934  ∀wal 1540  ∃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-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)