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Theorem sborv 1890
Description: Version of sbor 1954 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 1887 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  E. x ( x  =  y  /\  ( ph  \/  ps ) ) )
2 andi 818 . . . 4  |-  ( ( x  =  y  /\  ( ph  \/  ps )
)  <->  ( ( x  =  y  /\  ph )  \/  ( x  =  y  /\  ps )
) )
32exbii 1605 . . 3  |-  ( E. x ( x  =  y  /\  ( ph  \/  ps ) )  <->  E. x
( ( x  =  y  /\  ph )  \/  ( x  =  y  /\  ps ) ) )
4 19.43 1628 . . 3  |-  ( E. x ( ( x  =  y  /\  ph )  \/  ( x  =  y  /\  ps )
)  <->  ( E. x
( x  =  y  /\  ph )  \/ 
E. x ( x  =  y  /\  ps ) ) )
51, 3, 43bitri 206 . 2  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( E. x ( x  =  y  /\  ph )  \/  E. x
( x  =  y  /\  ps ) ) )
6 sb5 1887 . . 3  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
7 sb5 1887 . . 3  |-  ( [ y  /  x ] ps 
<->  E. x ( x  =  y  /\  ps ) )
86, 7orbi12i 764 . 2  |-  ( ( [ y  /  x ] ph  \/  [ y  /  x ] ps ) 
<->  ( E. x ( x  =  y  /\  ph )  \/  E. x
( x  =  y  /\  ps ) ) )
95, 8bitr4i 187 1  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    <-> wb 105    \/ wo 708   E.wex 1492   [wsb 1762
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-io 709  ax-5 1447  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534
This theorem depends on definitions:  df-bi 117  df-sb 1763
This theorem is referenced by:  sbor  1954
  Copyright terms: Public domain W3C validator