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Theorem rabn0r 3238
 Description: Non-empty restricted class abstraction. (Contributed by Jim Kingdon, 1-Aug-2018.)
Assertion
Ref Expression
rabn0r (x A φ → {x Aφ} ≠ ∅)

Proof of Theorem rabn0r
StepHypRef Expression
1 abn0r 3237 . 2 (x(x A φ) → {x ∣ (x A φ)} ≠ ∅)
2 df-rex 2306 . 2 (x A φx(x A φ))
3 df-rab 2309 . . 3 {x Aφ} = {x ∣ (x A φ)}
43neeq1i 2215 . 2 ({x Aφ} ≠ ∅ ↔ {x ∣ (x A φ)} ≠ ∅)
51, 2, 43imtr4i 190 1 (x A φ → {x Aφ} ≠ ∅)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1378   ∈ wcel 1390  {cab 2023   ≠ wne 2201  ∃wrex 2301  {crab 2304  ∅c0 3218 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-rex 2306  df-rab 2309  df-v 2553  df-dif 2914  df-nul 3219 This theorem is referenced by: (None)
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