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Theorem clelsb3 2577
Description: Substitution applied to an atomic wff (class version of elsb3 2283). (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 1769 . . 3  |-  F/ y  w  e.  A
21sbco2 2264 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  w ]
w  e.  A )
3 nfv 1769 . . . 4  |-  F/ w  y  e.  A
4 eleq1 2537 . . . 4  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
53, 4sbie 2257 . . 3  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
65sbbii 1812 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  y ] y  e.  A )
7 nfv 1769 . . 3  |-  F/ w  x  e.  A
8 eleq1 2537 . . 3  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
97, 8sbie 2257 . 2  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
102, 6, 93bitr3i 283 1  |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   [wsb 1805    e. wcel 1904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806  df-cleq 2464  df-clel 2467
This theorem is referenced by:  hblem  2579  cbvreu  3003  sbcel1v  3314  rmo3  3344  kmlem15  8612  iuninc  28253  measiuns  29113  ballotlemodife  29403  bj-nfcf  31595  sbcel1gvOLD  37318  ellimcabssub0  37794
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