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Theorem dfrab3 3207
Description: Alternate definition of restricted class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
dfrab3 {x Aφ} = (A ∩ {xφ})
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem dfrab3
StepHypRef Expression
1 df-rab 2309 . 2 {x Aφ} = {x ∣ (x A φ)}
2 inab 3199 . 2 ({xx A} ∩ {xφ}) = {x ∣ (x A φ)}
3 abid2 2155 . . 3 {xx A} = A
43ineq1i 3128 . 2 ({xx A} ∩ {xφ}) = (A ∩ {xφ})
51, 2, 43eqtr2i 2063 1 {x Aφ} = (A ∩ {xφ})
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242   wcel 1390  {cab 2023  {crab 2304  cin 2910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-in 2918
This theorem is referenced by:  notrab  3208  dfrab3ss  3209  dfif3  3337  dfse2  4641
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