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Theorem notab 3201
Description: A class builder defined by a negation. (Contributed by FL, 18-Sep-2010.)
Assertion
Ref Expression
notab {x ∣ ¬ φ} = (V ∖ {xφ})

Proof of Theorem notab
StepHypRef Expression
1 df-rab 2309 . . 3 {x V ∣ ¬ φ} = {x ∣ (x V ¬ φ)}
2 rabab 2569 . . 3 {x V ∣ ¬ φ} = {x ∣ ¬ φ}
31, 2eqtr3i 2059 . 2 {x ∣ (x V ¬ φ)} = {x ∣ ¬ φ}
4 difab 3200 . . 3 ({xx V} ∖ {xφ}) = {x ∣ (x V ¬ φ)}
5 abid2 2155 . . . 4 {xx V} = V
65difeq1i 3052 . . 3 ({xx V} ∖ {xφ}) = (V ∖ {xφ})
74, 6eqtr3i 2059 . 2 {x ∣ (x V ¬ φ)} = (V ∖ {xφ})
83, 7eqtr3i 2059 1 {x ∣ ¬ φ} = (V ∖ {xφ})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3   wa 97   = wceq 1242   wcel 1390  {cab 2023  {crab 2304  Vcvv 2551  cdif 2908
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-dif 2914
This theorem is referenced by:  dfif3  3337
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