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Theorem dfrab3 3860
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 2904 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 inab 3853 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴𝜑)}
3 abid2 2731 . . 3 {𝑥𝑥𝐴} = 𝐴
43ineq1i 3771 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑})
51, 2, 43eqtr2i 2637 1 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1474  wcel 1976  {cab 2595  {crab 2899  cin 3538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-rab 2904  df-v 3174  df-in 3546
This theorem is referenced by:  dfrab2  3861  notrab  3862  dfrab3ss  3863  dfif3  4049  dffr3  5403  dfse2  5404  tz6.26  5613  rabfi  8047  dfsup2  8210  ressmplbas2  19224  clsocv  22801  hasheuni  29267  bj-inrab3  31900  hashnzfz  37324
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