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Theorem rabeq2i 2684
Description: Inference from equality of a class variable and a restricted class abstraction. (Contributed by NM, 16-Feb-2004.)
Hypothesis
Ref Expression
rabeq2i.1  |-  A  =  { x  e.  B  |  ph }
Assertion
Ref Expression
rabeq2i  |-  ( x  e.  A  <->  ( x  e.  B  /\  ph )
)

Proof of Theorem rabeq2i
StepHypRef Expression
1 rabeq2i.1 . . 3  |-  A  =  { x  e.  B  |  ph }
21eleq2i 2207 . 2  |-  ( x  e.  A  <->  x  e.  { x  e.  B  |  ph } )
3 rabid 2607 . 2  |-  ( x  e.  { x  e.  B  |  ph }  <->  ( x  e.  B  /\  ph ) )
42, 3bitri 183 1  |-  ( x  e.  A  <->  ( x  e.  B  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1332    e. wcel 1481   {crab 2421
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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-rab 2426
This theorem is referenced by:  tfis  4501  fvmptssdm  5509  suplocsrlempr  7635  suplocsrlem  7636
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