Theorem rabn0 3571

Description: Nonempty restricted class abstraction. (Contributed by NM, 29-Aug-1999.)

Assertion

Ref	Expression
rabn0	⊢ ({x ∈ A ∣ φ} ≠ ∅ ↔ ∃x ∈ A φ)

Proof of Theorem rabn0

Step	Hyp	Ref	Expression
1		abn0 3569	. 2 ⊢ ({x ∣ (x ∈ A ∧ φ)} ≠ ∅ ↔ ∃x(x ∈ A ∧ φ))
2		df-rab 2624	. . 3 ⊢ {x ∈ A ∣ φ} = {x ∣ (x ∈ A ∧ φ)}
3	2	neeq1i 2527	. 2 ⊢ ({x ∈ A ∣ φ} ≠ ∅ ↔ {x ∣ (x ∈ A ∧ φ)} ≠ ∅)
4		df-rex 2621	. 2 ⊢ (∃x ∈ A φ ↔ ∃x(x ∈ A ∧ φ))
5	1, 3, 4	3bitr4i 268	1 ⊢ ({x ∈ A ∣ φ} ≠ ∅ ↔ ∃x ∈ A φ)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  {cab 2339   ≠ wne 2517  ∃wrex 2616  {crab 2619  ∅c0 3551

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-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-rex 2621  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  rabeq0  3573