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Theorem elsb4 1895
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 ax-17 1460 . . . . 5  |-  ( z  e.  y  ->  A. w  z  e.  y )
2 elequ2 1642 . . . . 5  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
31, 2sbieh 1714 . . . 4  |-  ( [ y  /  w ]
z  e.  w  <->  z  e.  y )
43sbbii 1689 . . 3  |-  ( [ x  /  y ] [ y  /  w ] z  e.  w  <->  [ x  /  y ] z  e.  y )
5 ax-17 1460 . . . 4  |-  ( z  e.  w  ->  A. y 
z  e.  w )
65sbco2h 1880 . . 3  |-  ( [ x  /  y ] [ y  /  w ] z  e.  w  <->  [ x  /  w ]
z  e.  w )
74, 6bitr3i 184 . 2  |-  ( [ x  /  y ] z  e.  y  <->  [ x  /  w ] z  e.  w )
8 equsb1 1709 . . . 4  |-  [ x  /  w ] w  =  x
9 elequ2 1642 . . . . 5  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
109sbimi 1688 . . . 4  |-  ( [ x  /  w ]
w  =  x  ->  [ x  /  w ] ( z  e.  w  <->  z  e.  x
) )
118, 10ax-mp 7 . . 3  |-  [ x  /  w ] ( z  e.  w  <->  z  e.  x )
12 sbbi 1875 . . 3  |-  ( [ x  /  w ]
( z  e.  w  <->  z  e.  x )  <->  ( [
x  /  w ]
z  e.  w  <->  [ x  /  w ] z  e.  x ) )
1311, 12mpbi 143 . 2  |-  ( [ x  /  w ]
z  e.  w  <->  [ x  /  w ] z  e.  x )
14 ax-17 1460 . . 3  |-  ( z  e.  x  ->  A. w  z  e.  x )
1514sbh 1700 . 2  |-  ( [ x  /  w ]
z  e.  x  <->  z  e.  x )
167, 13, 153bitri 204 1  |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   [wsb 1686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  peano2  4344
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