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Theorem clelsb3 2244
Description: Substitution applied to an atomic wff (class version of elsb3 1951). (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 1508 . . 3  |-  F/ x  w  e.  A
21sbco2 1938 . 2  |-  ( [ y  /  x ] [ x  /  w ] w  e.  A  <->  [ y  /  w ]
w  e.  A )
3 nfv 1508 . . . 4  |-  F/ w  x  e.  A
4 eleq1 2202 . . . 4  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
53, 4sbie 1764 . . 3  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
65sbbii 1738 . 2  |-  ( [ y  /  x ] [ x  /  w ] w  e.  A  <->  [ y  /  x ]
x  e.  A )
7 nfv 1508 . . 3  |-  F/ w  y  e.  A
8 eleq1 2202 . . 3  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
97, 8sbie 1764 . 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 1480   [wsb 1735
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135
This theorem is referenced by:  hblem  2247  nfraldya  2469  nfrexdya  2470  cbvreu  2652  sbcel1v  2971  rmo3  3000  setindel  4453  elirr  4456  en2lp  4469  zfregfr  4488  tfi  4496  bdcriota  13095
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