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Theorem abf 3930
 Description: A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
Hypothesis
Ref Expression
abf.1 ¬ 𝜑
Assertion
Ref Expression
abf {𝑥𝜑} = ∅

Proof of Theorem abf
StepHypRef Expression
1 ab0 3905 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
2 abf.1 . 2 ¬ 𝜑
31, 2mpgbir 1717 1 {𝑥𝜑} = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475  {cab 2596  ∅c0 3874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-nul 3875 This theorem is referenced by:  csbprc  3932  csbprcOLD  3933  mpt20  6623  fi0  8209  meet0  16960  join0  16961  rusgra0edg  26482  pmapglb2xN  34076
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