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Theorem clelsb3 2222
Description: Substitution applied to an atomic wff (class version of elsb3 1929). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
clelsb3  |-  ( [ y  /  x ]
x  e.  A  <->  y  e.  A )
Distinct variable group:    x, A
Allowed substitution hint:    A( y)

Proof of Theorem clelsb3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1493 . . 3  |-  F/ x  w  e.  A
21sbco2 1916 . 2  |-  ( [ y  /  x ] [ x  /  w ] w  e.  A  <->  [ y  /  w ]
w  e.  A )
3 nfv 1493 . . . 4  |-  F/ w  x  e.  A
4 eleq1 2180 . . . 4  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
53, 4sbie 1749 . . 3  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
65sbbii 1723 . 2  |-  ( [ y  /  x ] [ x  /  w ] w  e.  A  <->  [ y  /  x ]
x  e.  A )
7 nfv 1493 . . 3  |-  F/ w  y  e.  A
8 eleq1 2180 . . 3  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
97, 8sbie 1749 . 2  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
102, 6, 93bitr3i 209 1  |-  ( [ y  /  x ]
x  e.  A  <->  y  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    e. wcel 1465   [wsb 1720
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-cleq 2110  df-clel 2113
This theorem is referenced by:  hblem  2225  nfraldya  2446  nfrexdya  2447  cbvreu  2629  sbcel1v  2943  rmo3  2972  setindel  4423  elirr  4426  en2lp  4439  zfregfr  4458  tfi  4466  bdcriota  13008
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