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Theorem clelsb3 2544
Description: Substitution applied to an atomic wff (class version of elsb3 2185). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
clelsb3  |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
Distinct variable group:    y, A
Allowed substitution hint:    A( x)

Proof of Theorem clelsb3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1630 . . 3  |-  F/ y  w  e.  A
21sbco2 2167 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  w ]
w  e.  A )
3 nfv 1630 . . . 4  |-  F/ w  y  e.  A
4 eleq1 2502 . . . 4  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
53, 4sbie 2154 . . 3  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
65sbbii 1667 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  y ] y  e.  A )
7 nfv 1630 . . 3  |-  F/ w  x  e.  A
8 eleq1 2502 . . 3  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
97, 8sbie 2154 . 2  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
102, 6, 93bitr3i 268 1  |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   [wsb 1659    e. wcel 1727
This theorem is referenced by:  hblem  2546  cbvreu  2936  sbcel1v  3230  sbcel1gvOLD  3231  rmo3  3264  kmlem15  8075  iuninc  24042  measiuns  24602  ballotlemodife  24786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-cleq 2435  df-clel 2438
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