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Theorem clelab 2558
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 2425 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
21anbi2i 677 . . 3  |-  ( ( y  =  A  /\  y  e.  { x  |  ph } )  <->  ( y  =  A  /\  [ y  /  x ] ph ) )
32exbii 1593 . 2  |-  ( E. y ( y  =  A  /\  y  e. 
{ x  |  ph } )  <->  E. y
( y  =  A  /\  [ y  /  x ] ph ) )
4 df-clel 2434 . 2  |-  ( A  e.  { x  | 
ph }  <->  E. y
( y  =  A  /\  y  e.  {
x  |  ph }
) )
5 nfv 1630 . . 3  |-  F/ y ( x  =  A  /\  ph )
6 nfv 1630 . . . 4  |-  F/ x  y  =  A
7 nfs1v 2184 . . . 4  |-  F/ x [ y  /  x ] ph
86, 7nfan 1847 . . 3  |-  F/ x
( y  =  A  /\  [ y  /  x ] ph )
9 eqeq1 2444 . . . 4  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
10 sbequ12 1945 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
119, 10anbi12d 693 . . 3  |-  ( x  =  y  ->  (
( x  =  A  /\  ph )  <->  ( y  =  A  /\  [ y  /  x ] ph ) ) )
125, 8, 11cbvex 1984 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  A  /\  [ y  /  x ] ph ) )
133, 4, 123bitr4i 270 1  |-  ( A  e.  { x  | 
ph }  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551    = wceq 1653   [wsb 1659    e. wcel 1726   {cab 2424
This theorem is referenced by:  elrabi  3092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434
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