Theorem abn0OLD 4282

Description: Obsolete version of abn0 4281 as of 30-Aug-2024. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
abn0OLD	⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)

Proof of Theorem abn0OLD

Step	Hyp	Ref	Expression
1		nfab1 2899	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
2	1	n0f 4243	. 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2718	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	3	exbii 1855	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
5	2, 4	bitri 278	1 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 209  ∃wex 1787   ∈ wcel 2112  {cab 2714   ≠ wne 2932  ∅c0 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-dif 3856  df-nul 4224

This theorem is referenced by: (None)