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Theorem sborv 1883
Description: Version of sbor 1947 where  x and  y are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
Assertion
Ref Expression
sborv  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem sborv
StepHypRef Expression
1 sb5 1880 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  E. x ( x  =  y  /\  ( ph  \/  ps ) ) )
2 andi 813 . . . 4  |-  ( ( x  =  y  /\  ( ph  \/  ps )
)  <->  ( ( x  =  y  /\  ph )  \/  ( x  =  y  /\  ps )
) )
32exbii 1598 . . 3  |-  ( E. x ( x  =  y  /\  ( ph  \/  ps ) )  <->  E. x
( ( x  =  y  /\  ph )  \/  ( x  =  y  /\  ps ) ) )
4 19.43 1621 . . 3  |-  ( E. x ( ( x  =  y  /\  ph )  \/  ( x  =  y  /\  ps )
)  <->  ( E. x
( x  =  y  /\  ph )  \/ 
E. x ( x  =  y  /\  ps ) ) )
51, 3, 43bitri 205 . 2  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( E. x ( x  =  y  /\  ph )  \/  E. x
( x  =  y  /\  ps ) ) )
6 sb5 1880 . . 3  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
7 sb5 1880 . . 3  |-  ( [ y  /  x ] ps 
<->  E. x ( x  =  y  /\  ps ) )
86, 7orbi12i 759 . 2  |-  ( ( [ y  /  x ] ph  \/  [ y  /  x ] ps ) 
<->  ( E. x ( x  =  y  /\  ph )  \/  E. x
( x  =  y  /\  ps ) ) )
95, 8bitr4i 186 1  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104    \/ wo 703   E.wex 1485   [wsb 1755
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-io 704  ax-5 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-sb 1756
This theorem is referenced by:  sbor  1947
  Copyright terms: Public domain W3C validator