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Theorem rabid 2580
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 2399 . 2  |-  { x  e.  A  |  ph }  =  { x  |  ( x  e.  A  /\  ph ) }
21abeq2i 2225 1  |-  ( x  e.  { x  e.  A  |  ph }  <->  ( x  e.  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    e. wcel 1463   {crab 2394
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-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-rab 2399
This theorem is referenced by:  rabeq2i  2654  rabn0m  3356  repizf2lem  4045  rabxfrd  4350  onintrab2im  4394  tfis  4457  imasnopn  12310
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