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Theorem clelsb3 2192
Description: Substitution applied to an atomic wff (class version of elsb3 1900). (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 1466 . . 3  |-  F/ y  w  e.  A
21sbco2 1887 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  w ]
w  e.  A )
3 nfv 1466 . . . 4  |-  F/ w  y  e.  A
4 eleq1 2150 . . . 4  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
53, 4sbie 1721 . . 3  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
65sbbii 1695 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  y ] y  e.  A )
7 nfv 1466 . . 3  |-  F/ w  x  e.  A
8 eleq1 2150 . . 3  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
97, 8sbie 1721 . 2  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
102, 6, 93bitr3i 208 1  |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1438   [wsb 1692
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-cleq 2081  df-clel 2084
This theorem is referenced by:  hblem  2195  nfraldya  2412  nfrexdya  2413  cbvreu  2588  sbcel1v  2901  rmo3  2930  setindel  4354  elirr  4357  en2lp  4370  zfregfr  4389  tfi  4397  bdcriota  11774
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