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Theorem notab 3310
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 2397 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
2 rabab 2676 . . 3 {𝑥 ∈ V ∣ ¬ 𝜑} = {𝑥 ∣ ¬ 𝜑}
31, 2eqtr3i 2135 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = {𝑥 ∣ ¬ 𝜑}
4 difab 3309 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)}
5 abid2 2233 . . . 4 {𝑥𝑥 ∈ V} = V
65difeq1i 3154 . . 3 ({𝑥𝑥 ∈ V} ∖ {𝑥𝜑}) = (V ∖ {𝑥𝜑})
74, 6eqtr3i 2135 . 2 {𝑥 ∣ (𝑥 ∈ V ∧ ¬ 𝜑)} = (V ∖ {𝑥𝜑})
83, 7eqtr3i 2135 1 {𝑥 ∣ ¬ 𝜑} = (V ∖ {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wa 103   = wceq 1312  wcel 1461  {cab 2099  {crab 2392  Vcvv 2655  cdif 3032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rab 2397  df-v 2657  df-dif 3037
This theorem is referenced by:  dfif3  3451
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