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Theorem elsb4 1953
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  |-  ( [ y  /  x ]
z  e.  x  <->  z  e.  y )
Distinct variable group:    x, z

Proof of Theorem elsb4
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 ax-17 1507 . . . . 5  |-  ( z  e.  x  ->  A. w  z  e.  x )
2 elequ2 1692 . . . . 5  |-  ( w  =  x  ->  (
z  e.  w  <->  z  e.  x ) )
31, 2sbieh 1764 . . . 4  |-  ( [ x  /  w ]
z  e.  w  <->  z  e.  x )
43sbbii 1739 . . 3  |-  ( [ y  /  x ] [ x  /  w ] z  e.  w  <->  [ y  /  x ]
z  e.  x )
5 ax-17 1507 . . . 4  |-  ( z  e.  w  ->  A. x  z  e.  w )
65sbco2h 1938 . . 3  |-  ( [ y  /  x ] [ x  /  w ] z  e.  w  <->  [ y  /  w ]
z  e.  w )
74, 6bitr3i 185 . 2  |-  ( [ y  /  x ]
z  e.  x  <->  [ y  /  w ] z  e.  w )
8 equsb1 1759 . . . 4  |-  [ y  /  w ] w  =  y
9 elequ2 1692 . . . . 5  |-  ( w  =  y  ->  (
z  e.  w  <->  z  e.  y ) )
109sbimi 1738 . . . 4  |-  ( [ y  /  w ]
w  =  y  ->  [ y  /  w ] ( z  e.  w  <->  z  e.  y ) )
118, 10ax-mp 5 . . 3  |-  [ y  /  w ] ( z  e.  w  <->  z  e.  y )
12 sbbi 1933 . . 3  |-  ( [ y  /  w ]
( z  e.  w  <->  z  e.  y )  <->  ( [
y  /  w ]
z  e.  w  <->  [ y  /  w ] z  e.  y ) )
1311, 12mpbi 144 . 2  |-  ( [ y  /  w ]
z  e.  w  <->  [ y  /  w ] z  e.  y )
14 ax-17 1507 . . 3  |-  ( z  e.  y  ->  A. w  z  e.  y )
1514sbh 1750 . 2  |-  ( [ y  /  w ]
z  e.  y  <->  z  e.  y )
167, 13, 153bitri 205 1  |-  ( [ y  /  x ]
z  e.  x  <->  z  e.  y )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1736
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737
This theorem is referenced by: (None)
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