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Theorem abn0 4101
Description: Nonempty class abstraction. See also ab0 4098. (Contributed by NM, 26-Dec-1996.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
abn0 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)

Proof of Theorem abn0
StepHypRef Expression
1 nfab1 2915 . . 3 𝑥{𝑥𝜑}
21n0f 4075 . 2 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥 𝑥 ∈ {𝑥𝜑})
3 abid 2759 . . 3 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
43exbii 1924 . 2 (∃𝑥 𝑥 ∈ {𝑥𝜑} ↔ ∃𝑥𝜑)
52, 4bitri 264 1 ({𝑥𝜑} ≠ ∅ ↔ ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wex 1852  wcel 2145  {cab 2757  wne 2943  c0 4063
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-nul 4064
This theorem is referenced by:  intexab  4953  iinexg  4955  relimasn  5629  inisegn0  5638  mapprc  8013  modom  8317  tz9.1c  8770  scott0  8913  scott0s  8915  cp  8918  karden  8922  acnrcl  9065  aceq3lem  9143  cff  9272  cff1  9282  cfss  9289  domtriomlem  9466  axdclem  9543  nqpr  10038  supadd  11193  supmul  11197  hashf1lem2  13442  hashf1  13443  mreiincl  16464  efgval  18337  efger  18338  birthdaylem3  24901  disjex  29743  disjexc  29744  mppsval  31807  mblfinlem3  33781  ismblfin  33783  itg2addnc  33796  sdclem1  33871  upbdrech  40036
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