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Theorem clelab 2403
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 2270 . . . 4  |-  ( y  e.  { x  | 
ph }  <->  [ y  /  x ] ph )
21anbi2i 675 . . 3  |-  ( ( y  =  A  /\  y  e.  { x  |  ph } )  <->  ( y  =  A  /\  [ y  /  x ] ph ) )
32exbii 1569 . 2  |-  ( E. y ( y  =  A  /\  y  e. 
{ x  |  ph } )  <->  E. y
( y  =  A  /\  [ y  /  x ] ph ) )
4 df-clel 2279 . 2  |-  ( A  e.  { x  | 
ph }  <->  E. y
( y  =  A  /\  y  e.  {
x  |  ph }
) )
5 nfv 1605 . . 3  |-  F/ y ( x  =  A  /\  ph )
6 nfv 1605 . . . 4  |-  F/ x  y  =  A
7 nfs1v 2045 . . . 4  |-  F/ x [ y  /  x ] ph
86, 7nfan 1771 . . 3  |-  F/ x
( y  =  A  /\  [ y  /  x ] ph )
9 eqeq1 2289 . . . 4  |-  ( x  =  y  ->  (
x  =  A  <->  y  =  A ) )
10 sbequ12 1860 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
119, 10anbi12d 691 . . 3  |-  ( x  =  y  ->  (
( x  =  A  /\  ph )  <->  ( y  =  A  /\  [ y  /  x ] ph ) ) )
125, 8, 11cbvex 1925 . 2  |-  ( E. x ( x  =  A  /\  ph )  <->  E. y ( y  =  A  /\  [ y  /  x ] ph ) )
133, 4, 123bitr4i 268 1  |-  ( A  e.  { x  | 
ph }  <->  E. x
( x  =  A  /\  ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623   [wsb 1629    e. wcel 1684   {cab 2269
This theorem is referenced by:  intopcoaconb  25540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279
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