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Theorem sb9 1928
Description: Commutation of quantification and substitution variables. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 23-Mar-2018.)
Assertion
Ref Expression
sb9  |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )

Proof of Theorem sb9
Dummy variable  w is distinct from all other variables.
StepHypRef Expression
1 sb9v 1927 . . 3  |-  ( A. y [ y  /  w ] [ w  /  x ] ph  <->  A. w [ w  /  y ] [
w  /  x ] ph )
2 sbcom 1922 . . . 4  |-  ( [ w  /  y ] [ w  /  x ] ph  <->  [ w  /  x ] [ w  /  y ] ph )
32albii 1427 . . 3  |-  ( A. w [ w  /  y ] [ w  /  x ] ph  <->  A. w [ w  /  x ] [ w  /  y ] ph )
4 sb9v 1927 . . 3  |-  ( A. w [ w  /  x ] [ w  /  y ] ph  <->  A. x [ x  /  w ] [ w  /  y ] ph )
51, 3, 43bitri 205 . 2  |-  ( A. y [ y  /  w ] [ w  /  x ] ph  <->  A. x [ x  /  w ] [ w  /  y ] ph )
6 ax-17 1487 . . . 4  |-  ( ph  ->  A. w ph )
76sbco2h 1911 . . 3  |-  ( [ y  /  w ] [ w  /  x ] ph  <->  [ y  /  x ] ph )
87albii 1427 . 2  |-  ( A. y [ y  /  w ] [ w  /  x ] ph  <->  A. y [ y  /  x ] ph )
96sbco2h 1911 . . 3  |-  ( [ x  /  w ] [ w  /  y ] ph  <->  [ x  /  y ] ph )
109albii 1427 . 2  |-  ( A. x [ x  /  w ] [ w  /  y ] ph  <->  A. x [ x  /  y ] ph )
115, 8, 103bitr3ri 210 1  |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   A.wal 1310   [wsb 1716
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496
This theorem depends on definitions:  df-bi 116  df-nf 1418  df-sb 1717
This theorem is referenced by:  sb9i  1929
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