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Theorem clelab 2526
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 2377 . 2  |-  ( A  e.  { x  | 
ph }  <->  E. y
( y  =  A  /\  y  e.  {
x  |  ph }
) )
2 nfv 1715 . . 3  |-  F/ y ( x  =  A  /\  ph )
3 nfv 1715 . . . 4  |-  F/ x  y  =  A
4 nfsab1 2371 . . . 4  |-  F/ x  y  e.  { x  |  ph }
53, 4nfan 1936 . . 3  |-  F/ x
( y  =  A  /\  y  e.  {
x  |  ph }
)
6 eqeq1 2386 . . . 4  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
7 sbequ12 2000 . . . . 5  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
8 df-clab 2368 . . . . 5  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
97, 8syl6bbr 263 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  y  e.  { x  |  ph } ) )
106, 9anbi12d 708 . . 3  |-  ( x  =  y  ->  (
( x  =  A  /\  ph )  <->  ( y  =  A  /\  y  e.  { x  |  ph } ) ) )
112, 5, 10cbvex 2029 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  A  /\  y  e. 
{ x  |  ph } ) )
121, 11bitr4i 252 1  |-  ( A  e.  { x  | 
ph }  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1399   E.wex 1620   [wsb 1747    e. wcel 1826   {cab 2367
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377
This theorem is referenced by:  elrabi  3179  bj-csbsnlem  34817  frege55c  38417
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