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Theorem dfrab3 4305
Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfrab3 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfrab3
StepHypRef Expression
1 df-rab 3429 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 inab 4295 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴𝜑)}
3 abid2 2867 . . 3 {𝑥𝑥𝐴} = 𝐴
43ineq1i 4204 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑})
51, 2, 43eqtr2i 2762 1 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1534  wcel 2099  {cab 2705  {crab 3428  cin 3944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3429  df-v 3472  df-in 3952
This theorem is referenced by:  dfrab2  4306  notrab  4307  dfrab3ss  4308  dfif3  4538  dffr3  6097  dfse2  6098  tz6.26OLD  6348  rabfi  9287  dfsup2  9461  ressmplbas2  21958  clsocv  25171  hasheuni  33698  bj-inrab3  36401  bj-reabeq  36500  hashnzfz  43751
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