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Theorem clelab 2266
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
clelab  |-  ( A  e.  { x  | 
ph }  <->  E. x
( x  =  A  /\  ph ) )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem clelab
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 df-clab 2127 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
21anbi2i 453 . . 3  |-  ( ( y  =  A  /\  y  e.  { x  |  ph } )  <->  ( y  =  A  /\  [ y  /  x ] ph ) )
32exbii 1585 . 2  |-  ( E. y ( y  =  A  /\  y  e. 
{ x  |  ph } )  <->  E. y
( y  =  A  /\  [ y  /  x ] ph ) )
4 df-clel 2136 . 2  |-  ( A  e.  { x  | 
ph }  <->  E. y
( y  =  A  /\  y  e.  {
x  |  ph }
) )
5 nfv 1509 . . 3  |-  F/ y ( x  =  A  /\  ph )
6 nfv 1509 . . . 4  |-  F/ x  y  =  A
7 nfs1v 1913 . . . 4  |-  F/ x [ y  /  x ] ph
86, 7nfan 1545 . . 3  |-  F/ x
( y  =  A  /\  [ y  /  x ] ph )
9 eqeq1 2147 . . . 4  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
10 sbequ12 1745 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
119, 10anbi12d 465 . . 3  |-  ( x  =  y  ->  (
( x  =  A  /\  ph )  <->  ( y  =  A  /\  [ y  /  x ] ph ) ) )
125, 8, 11cbvex 1730 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  A  /\  [ y  /  x ] ph ) )
133, 4, 123bitr4i 211 1  |-  ( A  e.  { x  | 
ph }  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    = wceq 1332   E.wex 1469    e. wcel 1481   [wsb 1736   {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-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:  elrabi  2840
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