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Theorem abn0OLD 4376
Description: Obsolete version of abn0 4375 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 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)

Proof of Theorem abn0OLD
StepHypRef Expression
1 nfab1 2899 . . 3 𝑥{𝑥𝜑}
21n0f 4337 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ {𝑥𝜑})
3 abid 2707 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
43exbii 1842 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
52, 4bitri 275 1 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1773  wcel 2098  {cab 2703  wne 2934  c0 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-dif 3946  df-nul 4318
This theorem is referenced by: (None)
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