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Theorem clelab 2212
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 2075 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
21anbi2i 445 . . 3  |-  ( ( y  =  A  /\  y  e.  { x  |  ph } )  <->  ( y  =  A  /\  [ y  /  x ] ph ) )
32exbii 1541 . 2  |-  ( E. y ( y  =  A  /\  y  e. 
{ x  |  ph } )  <->  E. y
( y  =  A  /\  [ y  /  x ] ph ) )
4 df-clel 2084 . 2  |-  ( A  e.  { x  | 
ph }  <->  E. y
( y  =  A  /\  y  e.  {
x  |  ph }
) )
5 nfv 1466 . . 3  |-  F/ y ( x  =  A  /\  ph )
6 nfv 1466 . . . 4  |-  F/ x  y  =  A
7 nfs1v 1863 . . . 4  |-  F/ x [ y  /  x ] ph
86, 7nfan 1502 . . 3  |-  F/ x
( y  =  A  /\  [ y  /  x ] ph )
9 eqeq1 2094 . . . 4  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
10 sbequ12 1701 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
119, 10anbi12d 457 . . 3  |-  ( x  =  y  ->  (
( x  =  A  /\  ph )  <->  ( y  =  A  /\  [ y  /  x ] ph ) ) )
125, 8, 11cbvex 1686 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  A  /\  [ y  /  x ] ph ) )
133, 4, 123bitr4i 210 1  |-  ( A  e.  { x  | 
ph }  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103    = wceq 1289   E.wex 1426    e. wcel 1438   [wsb 1692   {cab 2074
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084
This theorem is referenced by:  elrabi  2766
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