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Theorem clelsb3 2386
Description: Substitution applied to an atomic wff (class version of elsb3 2042). (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
Dummy variable  w is distinct from all other variables.
Allowed substitution hint:    A( x)

Proof of Theorem clelsb3
StepHypRef Expression
1 nfv 1606 . . 3  |-  F/ y  w  e.  A
21sbco2 2025 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  w ]
w  e.  A )
3 nfv 1606 . . . 4  |-  F/ w  y  e.  A
4 eleq1 2344 . . . 4  |-  ( w  =  y  ->  (
w  e.  A  <->  y  e.  A ) )
53, 4sbie 1983 . . 3  |-  ( [ y  /  w ]
w  e.  A  <->  y  e.  A )
65sbbii 1636 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  A  <->  [ x  /  y ] y  e.  A )
7 nfv 1606 . . 3  |-  F/ w  x  e.  A
8 eleq1 2344 . . 3  |-  ( w  =  x  ->  (
w  e.  A  <->  x  e.  A ) )
97, 8sbie 1983 . 2  |-  ( [ x  /  w ]
w  e.  A  <->  x  e.  A )
102, 6, 93bitr3i 268 1  |-  ( [ x  /  y ] y  e.  A  <->  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   [wsb 1631    e. wcel 1685
This theorem is referenced by:  hblem  2388  cbvreu  2763  sbcel1gv  3051  rmo3  3079  kmlem15  7785
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-cleq 2277  df-clel 2280
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