Theorem abn0OLD 4380

Description: Obsolete version of abn0 4379 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 2894	. . 3 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
2	1	n0f 4343	. 2 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑})
3		abid 2707	. . 3 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
4	3	exbii 1843	. 2 ⊢ (∃𝑥 𝑥 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥𝜑)
5	2, 4	bitri 274	1 ⊢ ({𝑥 ∣ 𝜑} ≠ ∅ ↔ ∃𝑥𝜑)

Syntax hints:   ↔ wb 205  ∃wex 1774   ∈ wcel 2099  {cab 2703   ≠ wne 2930  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-dif 3950  df-nul 4324

This theorem is referenced by: (None)