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Theorem clelsb3 2158
Description: Substitution applied to an atomic wff (class version of elsb3 1868). (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 1437 . . 3  |-  F/ y  w  e.  A
21sbco2 1855 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  w ]
w  e.  A )
3 nfv 1437 . . . 4  |-  F/ w  y  e.  A
4 eleq1 2116 . . . 4  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
53, 4sbie 1690 . . 3  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
65sbbii 1664 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  y ] y  e.  A )
7 nfv 1437 . . 3  |-  F/ w  x  e.  A
8 eleq1 2116 . . 3  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
97, 8sbie 1690 . 2  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
102, 6, 93bitr3i 203 1  |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 102    e. wcel 1409   [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-cleq 2049  df-clel 2052
This theorem is referenced by:  hblem  2161  nfraldya  2375  nfrexdya  2376  cbvreu  2548  sbcel1v  2848  rmo3  2877  setindel  4291  elirr  4294  en2lp  4306  zfregfr  4326  tfi  4333  bdcriota  10390
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