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Theorem clabel 2179
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 2052 . 2  |-  ( { x  |  ph }  e.  A  <->  E. y ( y  =  { x  | 
ph }  /\  y  e.  A ) )
2 abeq2 2162 . . . 4  |-  ( y  =  { x  | 
ph }  <->  A. x
( x  e.  y  <->  ph ) )
32anbi2ci 440 . . 3  |-  ( ( y  =  { x  |  ph }  /\  y  e.  A )  <->  ( y  e.  A  /\  A. x
( x  e.  y  <->  ph ) ) )
43exbii 1512 . 2  |-  ( E. y ( y  =  { x  |  ph }  /\  y  e.  A
)  <->  E. y ( y  e.  A  /\  A. x ( x  e.  y  <->  ph ) ) )
51, 4bitri 177 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 101    <-> wb 102   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052
This theorem is referenced by: (None)
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