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Theorem rabn0OLD 4094
 Description: Obsolete proof of rabn0 4093 as of 16-Jul-2021. (Contributed by NM, 29-Aug-1999.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rabn0OLD ({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)

Proof of Theorem rabn0OLD
StepHypRef Expression
1 abn0 4089 . 2 ({𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅ ↔ ∃𝑥(𝑥𝐴𝜑))
2 df-rab 3051 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
32neeq1i 2988 . 2 ({𝑥𝐴𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅)
4 df-rex 3048 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
51, 3, 43bitr4i 292 1 ({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196   ∧ wa 383  ∃wex 1845   ∈ wcel 2131  {cab 2738   ≠ wne 2924  ∃wrex 3043  {crab 3046  ∅c0 4050 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-rex 3048  df-rab 3051  df-v 3334  df-dif 3710  df-nul 4051 This theorem is referenced by:  rabeq0OLD  4095
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