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Theorem clelsb4 2185
Description: Substitution applied to an atomic wff (class version of elsb4 1895). (Contributed by Jim Kingdon, 22-Nov-2018.)
Assertion
Ref Expression
clelsb4  |-  ( [ x  /  y ] A  e.  y  <->  A  e.  x )
Distinct variable group:    y, A
Allowed substitution hint:    A( x)

Proof of Theorem clelsb4
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1462 . . 3  |-  F/ y  A  e.  w
21sbco2 1881 . 2  |-  ( [ x  /  y ] [ y  /  w ] A  e.  w  <->  [ x  /  w ] A  e.  w )
3 nfv 1462 . . . 4  |-  F/ w  A  e.  y
4 eleq2 2143 . . . 4  |-  ( w  =  y  ->  ( A  e.  w  <->  A  e.  y ) )
53, 4sbie 1715 . . 3  |-  ( [ y  /  w ] A  e.  w  <->  A  e.  y )
65sbbii 1689 . 2  |-  ( [ x  /  y ] [ y  /  w ] A  e.  w  <->  [ x  /  y ] A  e.  y )
7 nfv 1462 . . 3  |-  F/ w  A  e.  x
8 eleq2 2143 . . 3  |-  ( w  =  x  ->  ( A  e.  w  <->  A  e.  x ) )
97, 8sbie 1715 . 2  |-  ( [ x  /  w ] A  e.  w  <->  A  e.  x )
102, 6, 93bitr3i 208 1  |-  ( [ x  /  y ] A  e.  y  <->  A  e.  x )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    e. wcel 1434   [wsb 1686
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-cleq 2075  df-clel 2078
This theorem is referenced by:  peano1  4343  peano2  4344
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