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Theorem abf 4359
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 4336 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
2 abf.1 . 2 ¬ 𝜑
31, 2mpgbir 1793 1 {𝑥𝜑} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1530  {cab 2803  c0 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-dif 3942  df-nul 4295
This theorem is referenced by:  csbprc  4361  mpo0  7232  fi0  8876  meet0  17739  join0  17740  fmla0disjsuc  32529  0qs  35489  pmapglb2xN  36775
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