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Theorem sbnv 1810
Description: Version of sbn 1868 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 1808 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  A. x ( x  =  y  ->  -.  ph ) )
2 alinexa 1535 . . 3  |-  ( A. x ( x  =  y  ->  -.  ph )  <->  -. 
E. x ( x  =  y  /\  ph ) )
31, 2bitri 182 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  E. x
( x  =  y  /\  ph ) )
4 sb5 1809 . 2  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
53, 4xchbinxr 641 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283   E.wex 1422   [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-in1 577  ax-in2 578  ax-5 1377  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-tru 1288  df-fal 1291  df-sb 1687
This theorem is referenced by:  sbn  1868
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