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Theorem clelsb2 2293
Description: Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2166). (Contributed by Jim Kingdon, 22-Nov-2018.)
Assertion
Ref Expression
clelsb2  |-  ( [ y  /  x ] A  e.  x  <->  A  e.  y )
Distinct variable group:    x, A
Allowed substitution hint:    A( y)

Proof of Theorem clelsb2
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1538 . . 3  |-  F/ x  A  e.  w
21sbco2 1975 . 2  |-  ( [ y  /  x ] [ x  /  w ] A  e.  w  <->  [ y  /  w ] A  e.  w )
3 nfv 1538 . . . 4  |-  F/ w  A  e.  x
4 eleq2 2251 . . . 4  |-  ( w  =  x  ->  ( A  e.  w  <->  A  e.  x ) )
53, 4sbie 1801 . . 3  |-  ( [ x  /  w ] A  e.  w  <->  A  e.  x )
65sbbii 1775 . 2  |-  ( [ y  /  x ] [ x  /  w ] A  e.  w  <->  [ y  /  x ] A  e.  x )
7 nfv 1538 . . 3  |-  F/ w  A  e.  y
8 eleq2 2251 . . 3  |-  ( w  =  y  ->  ( A  e.  w  <->  A  e.  y ) )
97, 8sbie 1801 . 2  |-  ( [ y  /  w ] A  e.  w  <->  A  e.  y )
102, 6, 93bitr3i 210 1  |-  ( [ y  /  x ] A  e.  x  <->  A  e.  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 105   [wsb 1772    e. wcel 2158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-nf 1471  df-sb 1773  df-cleq 2180  df-clel 2183
This theorem is referenced by:  peano1  4605  peano2  4606
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