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Theorem sbnv 1844
Description: Version of sbn 1903 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 1842 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  A. x ( x  =  y  ->  -.  ph ) )
2 alinexa 1567 . . 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 1843 . 2  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
53, 4xchbinxr 657 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 1314   E.wex 1453   [wsb 1720
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 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-sb 1721
This theorem is referenced by:  sbn  1903
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