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Theorem clelsb3 2360
Description: Substitution applied to an atomic wff (class version of elsb3 2066). (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
StepHypRef Expression
1 nfv 1629 . . 3  |-  F/ y  w  e.  A
21sbco2 1981 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  w ]
w  e.  A )
3 nfv 1629 . . . 4  |-  F/ w  y  e.  A
4 eleq1 2318 . . . 4  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
53, 4sbie 1911 . . 3  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
65sbbii 1886 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  y ] y  e.  A )
7 nfv 1629 . . 3  |-  F/ w  x  e.  A
8 eleq1 2318 . . 3  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
97, 8sbie 1911 . 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    e. wcel 1621   [wsb 1883
This theorem is referenced by:  hblem  2362  cbvreu  2737  sbcel1gv  3025  rmo3  3053  kmlem15  7758
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-cleq 2251  df-clel 2254
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