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Theorem sbor 2247
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 2244 . . 3  |-  ( [ y  /  x ]
( -.  ph  ->  ps )  <->  ( [ y  /  x ]  -.  ph 
->  [ y  /  x ] ps ) )
2 sbn 2240 . . . 4  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
32imbi1i 332 . . 3  |-  ( ( [ y  /  x ]  -.  ph  ->  [ y  /  x ] ps ) 
<->  ( -.  [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
41, 3bitri 257 . 2  |-  ( [ y  /  x ]
( -.  ph  ->  ps )  <->  ( -.  [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
5 df-or 377 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
65sbbii 1812 . 2  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  [ y  /  x ] ( -.  ph  ->  ps ) )
7 df-or 377 . 2  |-  ( ( [ y  /  x ] ph  \/  [ y  /  x ] ps ) 
<->  ( -.  [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
84, 6, 73bitr4i 285 1  |-  ( [ y  /  x ]
( ph  \/  ps ) 
<->  ( [ y  /  x ] ph  \/  [
y  /  x ] ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by:  sbcor  3299  unab  3701  sbcorgOLD  36961
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