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Theorem rabn0r 3464
Description: Nonempty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
Assertion
Ref Expression
rabn0r (∃𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ≠ ∅)

Proof of Theorem rabn0r
StepHypRef Expression
1 abn0r 3462 . 2 (∃𝑥(𝑥𝐴𝜑) → {𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅)
2 df-rex 2474 . 2 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
3 df-rab 2477 . . 3 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
43neeq1i 2375 . 2 ({𝑥𝐴𝜑} ≠ ∅ ↔ {𝑥 ∣ (𝑥𝐴𝜑)} ≠ ∅)
51, 2, 43imtr4i 201 1 (∃𝑥𝐴 𝜑 → {𝑥𝐴𝜑} ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wex 1503  wcel 2160  {cab 2175  wne 2360  wrex 2469  {crab 2472  c0 3437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-nul 3438
This theorem is referenced by: (None)
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