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Theorem orbidi 898
Description: Disjunction distributes over the biconditional. An axiom of system DS in Vladimir Lifschitz, "On calculational proofs" (1998), http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.25.3384. (Contributed by NM, 8-Jan-2005.) (Proof shortened by Wolf Lammen, 4-Feb-2013.)
Assertion
Ref Expression
orbidi  |-  ( (
ph  \/  ( ps  <->  ch ) )  <->  ( ( ph  \/  ps )  <->  ( ph  \/  ch ) ) )

Proof of Theorem orbidi
StepHypRef Expression
1 pm5.74 235 . 2  |-  ( ( -.  ph  ->  ( ps  <->  ch ) )  <->  ( ( -.  ph  ->  ps )  <->  ( -.  ph  ->  ch )
) )
2 df-or 359 . 2  |-  ( (
ph  \/  ( ps  <->  ch ) )  <->  ( -.  ph 
->  ( ps  <->  ch )
) )
3 df-or 359 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
4 df-or 359 . . 3  |-  ( (
ph  \/  ch )  <->  ( -.  ph  ->  ch )
)
53, 4bibi12i 306 . 2  |-  ( ( ( ph  \/  ps ) 
<->  ( ph  \/  ch ) )  <->  ( ( -.  ph  ->  ps )  <->  ( -.  ph  ->  ch )
) )
61, 2, 53bitr4i 268 1  |-  ( (
ph  \/  ( ps  <->  ch ) )  <->  ( ( ph  \/  ps )  <->  ( ph  \/  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357
This theorem is referenced by:  pm5.7  900
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359
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