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Theorem dfrab3 4304
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 3427 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 inab 4294 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴𝜑)}
3 abid2 2865 . . 3 {𝑥𝑥𝐴} = 𝐴
43ineq1i 4203 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑})
51, 2, 43eqtr2i 2760 1 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1533  wcel 2098  {cab 2703  {crab 3426  cin 3942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-in 3950
This theorem is referenced by:  dfrab2  4305  notrab  4306  dfrab3ss  4307  dfif3  4537  dffr3  6091  dfse2  6092  tz6.26OLD  6342  rabfi  9268  dfsup2  9438  ressmplbas2  21920  clsocv  25129  hasheuni  33613  bj-inrab3  36316  bj-reabeq  36415  hashnzfz  43636
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