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Theorem clelsb4 2223
Description: Substitution applied to an atomic wff (class version of elsb4 1930). (Contributed by Jim Kingdon, 22-Nov-2018.)
Assertion
Ref Expression
clelsb4  |-  ( [ y  /  x ] A  e.  x  <->  A  e.  y )
Distinct variable group:    x, A
Allowed substitution hint:    A( y)

Proof of Theorem clelsb4
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1493 . . 3  |-  F/ x  A  e.  w
21sbco2 1916 . 2  |-  ( [ y  /  x ] [ x  /  w ] A  e.  w  <->  [ y  /  w ] A  e.  w )
3 nfv 1493 . . . 4  |-  F/ w  A  e.  x
4 eleq2 2181 . . . 4  |-  ( w  =  x  ->  ( A  e.  w  <->  A  e.  x ) )
53, 4sbie 1749 . . 3  |-  ( [ x  /  w ] A  e.  w  <->  A  e.  x )
65sbbii 1723 . 2  |-  ( [ y  /  x ] [ x  /  w ] A  e.  w  <->  [ y  /  x ] A  e.  x )
7 nfv 1493 . . 3  |-  F/ w  A  e.  y
8 eleq2 2181 . . 3  |-  ( w  =  y  ->  ( A  e.  w  <->  A  e.  y ) )
97, 8sbie 1749 . 2  |-  ( [ y  /  w ] A  e.  w  <->  A  e.  y )
102, 6, 93bitr3i 209 1  |-  ( [ y  /  x ] A  e.  x  <->  A  e.  y )
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:  peano1  4478  peano2  4479
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