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Theorem dfrab2 4311
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 4310 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
2 incom 4202 . 2 (𝐴 ∩ {𝑥𝜑}) = ({𝑥𝜑} ∩ 𝐴)
31, 2eqtri 2761 1 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  {cab 2710  {crab 3433  cin 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-in 3956
This theorem is referenced by:  dfpred3  6312  predres  6341  lubdm  18304  glbdm  18317  psrbagsn  21624  ismbl  25043  lrrecse  27426  lrrecpred  27428  eulerpartgbij  33371  orvcval4  33459  fvline2  35118  abeqin  37120  rabdif  41032  nznngen  43075
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