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Theorem elsb4 2056
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 1609 . . 3  |-  F/ y  z  e.  w
21sbco2 2039 . 2  |-  ( [ x  /  y ] [ y  /  w ] z  e.  w  <->  [ x  /  w ]
z  e.  w )
3 nfv 1609 . . . 4  |-  F/ w  z  e.  y
4 elequ2 1701 . . . 4  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
53, 4sbie 1991 . . 3  |-  ( [ y  /  w ]
z  e.  w  <->  z  e.  y )
65sbbii 1643 . 2  |-  ( [ x  /  y ] [ y  /  w ] z  e.  w  <->  [ x  /  y ] z  e.  y )
7 nfv 1609 . . 3  |-  F/ w  z  e.  x
8 elequ2 1701 . . 3  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
97, 8sbie 1991 . 2  |-  ( [ x  /  w ]
z  e.  w  <->  z  e.  x )
102, 6, 93bitr3i 266 1  |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1638
This theorem is referenced by:  nfnid  4220
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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