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Theorem sbnv 1861
Description: Version of sbn 1926 where  x and  y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
Assertion
Ref Expression
sbnv  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem sbnv
StepHypRef Expression
1 sb6 1859 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  A. x ( x  =  y  ->  -.  ph ) )
2 alinexa 1583 . . 3  |-  ( A. x ( x  =  y  ->  -.  ph )  <->  -. 
E. x ( x  =  y  /\  ph ) )
31, 2bitri 183 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  E. x
( x  =  y  /\  ph ) )
4 sb5 1860 . 2  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
53, 4xchbinxr 673 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1330   E.wex 1469   [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-in1 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-11 1485  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-sb 1737
This theorem is referenced by:  sbn  1926
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