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Theorem dfrab2 3862
 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 3861 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
2 incom 3767 . 2 (𝐴 ∩ {𝑥𝜑}) = ({𝑥𝜑} ∩ 𝐴)
31, 2eqtri 2632 1 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1475  {cab 2596  {crab 2900   ∩ cin 3539 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-in 3547 This theorem is referenced by:  dfpred3  5607  lubdm  16802  glbdm  16815  psrbagsn  19316  ismbl  23101  eulerpartgbij  29761  orvcval4  29849  fvline2  31423  nznngen  37537
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