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Theorem abf 3584
 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 {x φ} =

Proof of Theorem abf
StepHypRef Expression
1 abf.1 . . . 4 ¬ φ
21pm2.21i 123 . . 3 (φx )
32abssi 3341 . 2 {x φ}
4 ss0 3581 . 2 ({x φ} → {x φ} = )
53, 4ax-mp 5 1 {x φ} =
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  {cab 2339   ⊆ wss 3257  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551 This theorem is referenced by:  evenodddisj  4516
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