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Theorem dfrab2 4302
Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.) (Proof shortened by BJ, 22-Apr-2019.)
Assertion
Ref Expression
dfrab2 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfrab2
StepHypRef Expression
1 dfrab3 4301 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
2 incom 4193 . 2 (𝐴 ∩ {𝑥𝜑}) = ({𝑥𝜑} ∩ 𝐴)
31, 2eqtri 2752 1 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  {cab 2701  {crab 3424  cin 3939
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 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-in 3947
This theorem is referenced by:  dfpred3  6301  predres  6330  lubdm  18305  glbdm  18318  psrbagsn  21933  ismbl  25376  lrrecse  27774  lrrecpred  27776  eulerpartgbij  33826  orvcval4  33914  fvline2  35579  abeqin  37576  rabdif  41491  nznngen  43530
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