Theorem vtocl2 2911

Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
vtocl2.1	⊢ A ∈ V
vtocl2.2	⊢ B ∈ V
vtocl2.3	⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
vtocl2.4	⊢ φ

Assertion

Ref	Expression
vtocl2	⊢ ψ

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

Allowed substitution hints:   φ(x,y)

Proof of Theorem vtocl2

Step	Hyp	Ref	Expression
1		vtocl2.1	. . . . . 6 ⊢ A ∈ V
2	1	isseti 2866	. . . . 5 ⊢ ∃x x = A
3		vtocl2.2	. . . . . 6 ⊢ B ∈ V
4	3	isseti 2866	. . . . 5 ⊢ ∃y y = B
5		eeanv 1913	. . . . . 6 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
6		vtocl2.3	. . . . . . . 8 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
7	6	biimpd 198	. . . . . . 7 ⊢ ((x = A ∧ y = B) → (φ → ψ))
8	7	2eximi 1577	. . . . . 6 ⊢ (∃x∃y(x = A ∧ y = B) → ∃x∃y(φ → ψ))
9	5, 8	sylbir 204	. . . . 5 ⊢ ((∃x x = A ∧ ∃y y = B) → ∃x∃y(φ → ψ))
10	2, 4, 9	mp2an 653	. . . 4 ⊢ ∃x∃y(φ → ψ)
11		19.36v 1896	. . . . 5 ⊢ (∃y(φ → ψ) ↔ (∀yφ → ψ))
12	11	exbii 1582	. . . 4 ⊢ (∃x∃y(φ → ψ) ↔ ∃x(∀yφ → ψ))
13	10, 12	mpbi 199	. . 3 ⊢ ∃x(∀yφ → ψ)
14	13	19.36aiv 1897	. 2 ⊢ (∀x∀yφ → ψ)
15		vtocl2.4	. . 3 ⊢ φ
16	15	ax-gen 1546	. 2 ⊢ ∀yφ
17	14, 16	mpg 1548	1 ⊢ ψ

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃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-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:  caovord  5630