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Theorem notrab 4321
Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.)
Assertion
Ref Expression
notrab (𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem notrab
StepHypRef Expression
1 difab 4309 . 2 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
2 difin 4271 . . 3 (𝐴 ∖ (𝐴 ∩ {𝑥𝜑})) = (𝐴 ∖ {𝑥𝜑})
3 dfrab3 4318 . . . 4 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
43difeq2i 4122 . . 3 (𝐴 ∖ {𝑥𝐴𝜑}) = (𝐴 ∖ (𝐴 ∩ {𝑥𝜑}))
5 abid2 2878 . . . 4 {𝑥𝑥𝐴} = 𝐴
65difeq1i 4121 . . 3 ({𝑥𝑥𝐴} ∖ {𝑥𝜑}) = (𝐴 ∖ {𝑥𝜑})
72, 4, 63eqtr4i 2774 . 2 (𝐴 ∖ {𝑥𝐴𝜑}) = ({𝑥𝑥𝐴} ∖ {𝑥𝜑})
8 df-rab 3436 . 2 {𝑥𝐴 ∣ ¬ 𝜑} = {𝑥 ∣ (𝑥𝐴 ∧ ¬ 𝜑)}
91, 7, 83eqtr4i 2774 1 (𝐴 ∖ {𝑥𝐴𝜑}) = {𝑥𝐴 ∣ ¬ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1539  wcel 2107  {cab 2713  {crab 3435  cdif 3947  cin 3949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-in 3957
This theorem is referenced by:  rlimrege0  15616  mhpmulcl  22154  ordtcld1  23206  ordtcld2  23207  lhop1lem  26053  rpvmasumlem  27532  finsumvtxdg2ssteplem1  29564  frgrwopreglem3  30334  zarcls  33874  hasheuni  34087  braew  34244  satfv1  35369
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