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Theorem sb9v 1896
Description: Like sb9 1897 but with a distinct variable constraint between  x and  y. (Contributed by Jim Kingdon, 28-Feb-2018.)
Assertion
Ref Expression
sb9v  |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sb9v
StepHypRef Expression
1 hbs1 1856 . 2  |-  ( [ x  /  y ]
ph  ->  A. y [ x  /  y ] ph )
2 hbs1 1856 . 2  |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
3 sbequ12 1695 . . . 4  |-  ( y  =  x  ->  ( ph 
<->  [ x  /  y ] ph ) )
43equcoms 1635 . . 3  |-  ( x  =  y  ->  ( ph 
<->  [ x  /  y ] ph ) )
5 sbequ12 1695 . . 3  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
64, 5bitr3d 188 . 2  |-  ( x  =  y  ->  ( [ x  /  y ] ph  <->  [ y  /  x ] ph ) )
71, 2, 6cbvalh 1677 1  |-  ( A. x [ x  /  y ] ph  <->  A. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 103   A.wal 1283   [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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687
This theorem is referenced by:  sb9  1897
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