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Theorem clabel 2404
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 2279 . 2  |-  ( { x  |  ph }  e.  A  <->  E. y ( y  =  { x  | 
ph }  /\  y  e.  A ) )
2 abeq2 2388 . . . 4  |-  ( y  =  { x  | 
ph }  <->  A. x
( x  e.  y  <->  ph ) )
32anbi2ci 677 . . 3  |-  ( ( y  =  { x  |  ph }  /\  y  e.  A )  <->  ( y  e.  A  /\  A. x
( x  e.  y  <->  ph ) ) )
43exbii 1569 . 2  |-  ( E. y ( y  =  { x  |  ph }  /\  y  e.  A
)  <->  E. y ( y  e.  A  /\  A. x ( x  e.  y  <->  ph ) ) )
51, 4bitri 240 1  |-  ( { x  |  ph }  e.  A  <->  E. y ( y  e.  A  /\  A. x ( x  e.  y  <->  ph ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269
This theorem is referenced by:  grothprimlem  8455
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279
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