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Theorem elsb3 2042
Description: Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
Assertion
Ref Expression
elsb3  |-  ( [ x  /  y ] y  e.  z  <->  x  e.  z )
Distinct variable group:    y, z

Proof of Theorem elsb3
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 nfv 1605 . . 3  |-  F/ y  w  e.  z
21sbco2 2026 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  z  <->  [ x  /  w ]
w  e.  z )
3 nfv 1605 . . . 4  |-  F/ w  y  e.  z
4 elequ1 1687 . . . 4  |-  ( w  =  y  ->  (
w  e.  z  <->  y  e.  z ) )
53, 4sbie 1978 . . 3  |-  ( [ y  /  w ]
w  e.  z  <->  y  e.  z )
65sbbii 1634 . 2  |-  ( [ x  /  y ] [ y  /  w ] w  e.  z  <->  [ x  /  y ] y  e.  z )
7 nfv 1605 . . 3  |-  F/ w  x  e.  z
8 elequ1 1687 . . 3  |-  ( w  =  x  ->  (
w  e.  z  <->  x  e.  z ) )
97, 8sbie 1978 . 2  |-  ( [ x  /  w ]
w  e.  z  <->  x  e.  z )
102, 6, 93bitr3i 266 1  |-  ( [ x  /  y ] y  e.  z  <->  x  e.  z )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1629
This theorem is referenced by:  cvjust  2278
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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