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Theorem notab 3525
 Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
Assertion
Ref Expression
notab {x ¬ φ} = (V {x φ})

Proof of Theorem notab
StepHypRef Expression
1 df-rab 2623 . . 3 {x V ¬ φ} = {x (x V ¬ φ)}
2 rabab 2876 . . 3 {x V ¬ φ} = {x ¬ φ}
31, 2eqtr3i 2375 . 2 {x (x V ¬ φ)} = {x ¬ φ}
4 difab 3523 . . 3 ({x x V} {x φ}) = {x (x V ¬ φ)}
5 abid2 2470 . . . 4 {x x V} = V
65difeq1i 3381 . . 3 ({x x V} {x φ}) = (V {x φ})
74, 6eqtr3i 2375 . 2 {x (x V ¬ φ)} = (V {x φ})
83, 7eqtr3i 2375 1 {x ¬ φ} = (V {x φ})
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618  Vcvv 2859   ∖ cdif 3206 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-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215 This theorem is referenced by:  dfif3  3672
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