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Theorem rabid 2500
Description: An "identity" law of concretion for restricted abstraction. Special case of Definition 2.1 of [Quine] p. 16. (Contributed by NM, 9-Oct-2003.)
Assertion
Ref Expression
rabid  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )

Proof of Theorem rabid
StepHypRef Expression
1 df-rab 2330 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21abeq2i 2162 1  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    e. wcel 1407   {crab 2325
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1350  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-rab 2330
This theorem is referenced by:  rabeq2i  2569  rabn0m  3270  repizf2lem  3939  rabxfrd  4226  onintrab2im  4269  tfis  4331
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