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Theorem abf 4122
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 4098 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
2 abf.1 . 2 ¬ 𝜑
31, 2mpgbir 1874 1 {𝑥𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1631  {cab 2757  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 827  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-v 3353  df-dif 3726  df-nul 4064
This theorem is referenced by:  csbprc  4124  csbprcOLD  4125  mpt20  6871  fi0  8481  meet0  17344  join0  17345  0qs  34470  pmapglb2xN  35576
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