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Theorem sborv 1786
Description: Version of sbor 1844 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 1783 . . 3  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  E. x ( x  =  y  /\  ( ph  \/  ps ) ) )
2 andi 742 . . . 4  |-  ( ( x  =  y  /\  ( ph  \/  ps )
)  <->  ( ( x  =  y  /\  ph )  \/  ( x  =  y  /\  ps )
) )
32exbii 1512 . . 3  |-  ( E. x ( x  =  y  /\  ( ph  \/  ps ) )  <->  E. x
( ( x  =  y  /\  ph )  \/  ( x  =  y  /\  ps ) ) )
4 19.43 1535 . . 3  |-  ( E. x ( ( x  =  y  /\  ph )  \/  ( x  =  y  /\  ps )
)  <->  ( E. x
( x  =  y  /\  ph )  \/ 
E. x ( x  =  y  /\  ps ) ) )
51, 3, 43bitri 199 . 2  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( E. x ( x  =  y  /\  ph )  \/  E. x
( x  =  y  /\  ps ) ) )
6 sb5 1783 . . 3  |-  ( [ y  /  x ] ph 
<->  E. x ( x  =  y  /\  ph ) )
7 sb5 1783 . . 3  |-  ( [ y  /  x ] ps 
<->  E. x ( x  =  y  /\  ps ) )
86, 7orbi12i 691 . 2  |-  ( ( [ y  /  x ] ph  \/  [ y  /  x ] ps ) 
<->  ( E. x ( x  =  y  /\  ph )  \/  E. x
( x  =  y  /\  ps ) ) )
95, 8bitr4i 180 1  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    <-> wb 102    \/ wo 639   E.wex 1397   [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbor  1844
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