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Theorem dfrab2 4067
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 4066 . 2 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
2 incom 3967 . 2 (𝐴 ∩ {𝑥𝜑}) = ({𝑥𝜑} ∩ 𝐴)
31, 2eqtri 2787 1 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  {cab 2751  {crab 3059  cin 3731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-rab 3064  df-v 3352  df-in 3739
This theorem is referenced by:  dfpred3  5875  lubdm  17245  glbdm  17258  psrbagsn  19768  ismbl  23584  eulerpartgbij  30881  orvcval4  30970  fvline2  32697  abeqin  34452  nznngen  39189
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