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Theorem clabel 2267
Description: Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
Assertion
Ref Expression
clabel  |-  ( { x  |  ph }  e.  A  <->  E. y ( y  e.  A  /\  A. x ( x  e.  y  <->  ph ) ) )
Distinct variable groups:    y, A    ph, y    x, y
Allowed substitution hints:    ph( x)    A( x)

Proof of Theorem clabel
StepHypRef Expression
1 df-clel 2136 . 2  |-  ( { x  |  ph }  e.  A  <->  E. y ( y  =  { x  | 
ph }  /\  y  e.  A ) )
2 abeq2 2249 . . . 4  |-  ( y  =  { x  | 
ph }  <->  A. x
( x  e.  y  <->  ph ) )
32anbi2ci 455 . . 3  |-  ( ( y  =  { x  |  ph }  /\  y  e.  A )  <->  ( y  e.  A  /\  A. x
( x  e.  y  <->  ph ) ) )
43exbii 1585 . 2  |-  ( E. y ( y  =  { x  |  ph }  /\  y  e.  A
)  <->  E. y ( y  e.  A  /\  A. x ( x  e.  y  <->  ph ) ) )
51, 4bitri 183 1  |-  ( { x  |  ph }  e.  A  <->  E. y ( y  e.  A  /\  A. x ( x  e.  y  <->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   A.wal 1330    = wceq 1332   E.wex 1469    e. wcel 1481   {cab 2126
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-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136
This theorem is referenced by:  frecabcl  6300
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