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Theorem sbor 2140
Description: Logical OR inside and outside of substitution are equivalent. (Contributed by NM, 29-Sep-2002.)
Assertion
Ref Expression
sbor  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )

Proof of Theorem sbor
StepHypRef Expression
1 sbim 2137 . . 3  |-  ( [ y  /  x ]
( -.  ph  ->  ps )  <->  ( [ y  /  x ]  -.  ph 
->  [ y  /  x ] ps ) )
2 sbn 2132 . . . 4  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
32imbi1i 317 . . 3  |-  ( ( [ y  /  x ]  -.  ph  ->  [ y  /  x ] ps ) 
<->  ( -.  [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
41, 3bitri 242 . 2  |-  ( [ y  /  x ]
( -.  ph  ->  ps )  <->  ( -.  [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
5 df-or 361 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
65sbbii 1666 . 2  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  [ y  /  x ] ( -.  ph  ->  ps ) )
7 df-or 361 . 2  |-  ( ( [ y  /  x ] ph  \/  [ y  /  x ] ps ) 
<->  ( -.  [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
84, 6, 73bitr4i 270 1  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359   [wsb 1659
This theorem is referenced by:  sbcor  3207  sbcorg  3208  unab  3610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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