Theorem abn0 3569

Description: Nonempty class abstraction. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)

Assertion

Ref	Expression
abn0	⊢ ({x ∣ φ} ≠ ∅ ↔ ∃xφ)

Proof of Theorem abn0

Step	Hyp	Ref	Expression
1		nfab1 2492	. . 3 ⊢ Ⅎx{x ∣ φ}
2	1	n0f 3559	. 2 ⊢ ({x ∣ φ} ≠ ∅ ↔ ∃x x ∈ {x ∣ φ})
3		abid 2341	. . 3 ⊢ (x ∈ {x ∣ φ} ↔ φ)
4	3	exbii 1582	. 2 ⊢ (∃x x ∈ {x ∣ φ} ↔ ∃xφ)
5	2, 4	bitri 240	1 ⊢ ({x ∣ φ} ≠ ∅ ↔ ∃xφ)

Syntax hints:   ↔ wb 176  ∃wex 1541   ∈ wcel 1710  {cab 2339   ≠ wne 2517  ∅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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by:  ab0  3570  rabn0  3571  imasn  5019  frds  5936  mapprc  6005  map0b  6025  map0  6026