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Theorem dfrab2 4326
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 4325 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
2 incom 4217 . 2 (𝐴 ∩ {𝑥𝜑}) = ({𝑥𝜑} ∩ 𝐴)
31, 2eqtri 2763 1 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  {cab 2712  {crab 3433  cin 3962
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-in 3970
This theorem is referenced by:  rabdif  4327  dfpred3  6334  predres  6362  lubdm  18409  glbdm  18422  psrbagsn  22105  ismbl  25575  lrrecse  27990  lrrecpred  27992  eulerpartgbij  34354  orvcval4  34442  fvline2  36128  abeqin  38234  nznngen  44312
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