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Theorem sbnv 1900
Description: Version of sbn 1964 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 1898 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  A. x ( x  =  y  ->  -.  ph ) )
2 alinexa 1614 . . 3  |-  ( A. x ( x  =  y  ->  -.  ph )  <->  -. 
E. x ( x  =  y  /\  ph ) )
31, 2bitri 184 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  E. x
( x  =  y  /\  ph ) )
4 sb5 1899 . 2  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
53, 4xchbinxr 684 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105   A.wal 1362   E.wex 1503   [wsb 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-sb 1774
This theorem is referenced by:  sbn  1964
  Copyright terms: Public domain W3C validator