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Theorem rabeq2i 2760
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 2263 . 2  |-  ( x  e.  A  <->  x  e.  { x  e.  B  |  ph } )
3 rabid 2673 . 2  |-  ( x  e.  { x  e.  B  |  ph }  <->  ( x  e.  B  /\  ph ) )
42, 3bitri 184 1  |-  ( x  e.  A  <->  ( x  e.  B  /\  ph )
)
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    = wceq 1364    e. wcel 2167   {crab 2479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-ext 2178
This theorem depends on definitions:  df-bi 117  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-rab 2484
This theorem is referenced by:  tfis  4619  fvmptssdm  5646  suplocsrlempr  7872  suplocsrlem  7873
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