Theorem clabel 2475

Description: Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)

Assertion

Ref	Expression
clabel	⊢ ({x ∣ φ} ∈ A ↔ ∃y(y ∈ A ∧ ∀x(x ∈ y ↔ φ)))

Distinct variable groups:   y,A   φ,y   x,y

Allowed substitution hints:   φ(x)   A(x)

Proof of Theorem clabel

Step	Hyp	Ref	Expression
1		df-clel 2349	. 2 ⊢ ({x ∣ φ} ∈ A ↔ ∃y(y = {x ∣ φ} ∧ y ∈ A))
2		abeq2 2459	. . . 4 ⊢ (y = {x ∣ φ} ↔ ∀x(x ∈ y ↔ φ))
3	2	anbi2ci 677	. . 3 ⊢ ((y = {x ∣ φ} ∧ y ∈ A) ↔ (y ∈ A ∧ ∀x(x ∈ y ↔ φ)))
4	3	exbii 1582	. 2 ⊢ (∃y(y = {x ∣ φ} ∧ y ∈ A) ↔ ∃y(y ∈ A ∧ ∀x(x ∈ y ↔ φ)))
5	1, 4	bitri 240	1 ⊢ ({x ∣ φ} ∈ A ↔ ∃y(y ∈ A ∧ ∀x(x ∈ y ↔ φ)))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339

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-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349

This theorem is referenced by: (None)