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Theorem elsb4 2180
Description: Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb4  |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
Distinct variable group:    y, z

Proof of Theorem elsb4
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1629 . . 3  |-  F/ y  z  e.  w
21sbco2 2161 . 2  |-  ( [ x  /  y ] [ y  /  w ] z  e.  w  <->  [ x  /  w ]
z  e.  w )
3 nfv 1629 . . . 4  |-  F/ w  z  e.  y
4 elequ2 1730 . . . 4  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
53, 4sbie 2149 . . 3  |-  ( [ y  /  w ]
z  e.  w  <->  z  e.  y )
65sbbii 1665 . 2  |-  ( [ x  /  y ] [ y  /  w ] z  e.  w  <->  [ x  /  y ] z  e.  y )
7 nfv 1629 . . 3  |-  F/ w  z  e.  x
8 elequ2 1730 . . 3  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
97, 8sbie 2149 . 2  |-  ( [ x  /  w ]
z  e.  w  <->  z  e.  x )
102, 6, 93bitr3i 267 1  |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [wsb 1658
This theorem is referenced by:  nfnid  4393
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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