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Theorem clelsb3f 28109
Description: Substitution applied to an atomic wff (class version of elsb3 2262). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
Hypothesis
Ref Expression
clelsb3f.1  |-  F/_ y A
Assertion
Ref Expression
clelsb3f  |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )

Proof of Theorem clelsb3f
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 clelsb3f.1 . . . 4  |-  F/_ y A
21nfcri 2585 . . 3  |-  F/ y  w  e.  A
32sbco2 2243 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  w ]
w  e.  A )
4 nfv 1760 . . . 4  |-  F/ w  y  e.  A
5 eleq1 2516 . . . 4  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
64, 5sbie 2236 . . 3  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
76sbbii 1803 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  y ] y  e.  A )
8 nfv 1760 . . 3  |-  F/ w  x  e.  A
9 eleq1 2516 . . 3  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
108, 9sbie 2236 . 2  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
113, 7, 103bitr3i 279 1  |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188   [wsb 1796    e. wcel 1886   F/_wnfc 2578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1663  df-nf 1667  df-sb 1797  df-cleq 2443  df-clel 2446  df-nfc 2580
This theorem is referenced by:  rmo3f  28124  suppss2fOLD  28230  suppss2f  28231  fmptdF  28248  disjdsct  28276  esumpfinvalf  28890
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