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Theorem clelab 2587
Description: Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)
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-clel 2458 . 2  |-  ( A  e.  { x  | 
ph }  <->  E. y
( y  =  A  /\  y  e.  {
x  |  ph }
) )
2 nfv 1772 . . 3  |-  F/ y ( x  =  A  /\  ph )
3 nfv 1772 . . . 4  |-  F/ x  y  =  A
4 nfsab1 2452 . . . 4  |-  F/ x  y  e.  { x  |  ph }
53, 4nfan 2022 . . 3  |-  F/ x
( y  =  A  /\  y  e.  {
x  |  ph }
)
6 eqeq1 2466 . . . 4  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
7 sbequ12 2094 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
8 df-clab 2449 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
97, 8syl6bbr 271 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  y  e.  { x  |  ph } ) )
106, 9anbi12d 722 . . 3  |-  ( x  =  y  ->  (
( x  =  A  /\  ph )  <->  ( y  =  A  /\  y  e.  { x  |  ph } ) ) )
112, 5, 10cbvex 2126 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  A  /\  y  e. 
{ x  |  ph } ) )
121, 11bitr4i 260 1  |-  ( A  e.  { x  | 
ph }  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375    = wceq 1455   E.wex 1674   [wsb 1808    e. wcel 1898   {cab 2448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458
This theorem is referenced by:  elrabi  3205  bj-csbsnlem  31550  frege55c  36559
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