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Theorem dfrab2 3315
Description: Alternate definition of restricted class abstraction. (Contributed by NM, 20-Sep-2003.)
Assertion
Ref Expression
dfrab2 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfrab2
StepHypRef Expression
1 df-rab 2397 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 inab 3308 . . 3 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴𝜑)}
3 abid2 2233 . . . 4 {𝑥𝑥𝐴} = 𝐴
43ineq1i 3237 . . 3 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑})
52, 4eqtr3i 2135 . 2 {𝑥 ∣ (𝑥𝐴𝜑)} = (𝐴 ∩ {𝑥𝜑})
6 incom 3232 . 2 (𝐴 ∩ {𝑥𝜑}) = ({𝑥𝜑} ∩ 𝐴)
71, 5, 63eqtri 2137 1 {𝑥𝐴𝜑} = ({𝑥𝜑} ∩ 𝐴)
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1312  wcel 1461  {cab 2099  {crab 2392  cin 3034
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095
This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rab 2397  df-v 2657  df-in 3041
This theorem is referenced by:  minmax  10887  xrminmax  10920
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