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Theorem dfrab3 3320
Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfrab3 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfrab3
StepHypRef Expression
1 df-rab 2400 . 2 {𝑥𝐴𝜑} = {𝑥 ∣ (𝑥𝐴𝜑)}
2 inab 3312 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = {𝑥 ∣ (𝑥𝐴𝜑)}
3 abid2 2236 . . 3 {𝑥𝑥𝐴} = 𝐴
43ineq1i 3241 . 2 ({𝑥𝑥𝐴} ∩ {𝑥𝜑}) = (𝐴 ∩ {𝑥𝜑})
51, 2, 43eqtr2i 2142 1 {𝑥𝐴𝜑} = (𝐴 ∩ {𝑥𝜑})
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1314  wcel 1463  {cab 2101  {crab 2395  cin 3038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-in 3045
This theorem is referenced by:  notrab  3321  dfrab3ss  3322  dfif3  3455  dfse2  4880
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