MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  orimdi Structured version   Unicode version

Theorem orimdi 821
Description: Disjunction distributes over implication. (Contributed by Wolf Lammen, 5-Jan-2013.)
Assertion
Ref Expression
orimdi  |-  ( (
ph  \/  ( ps  ->  ch ) )  <->  ( ( ph  \/  ps )  -> 
( ph  \/  ch ) ) )

Proof of Theorem orimdi
StepHypRef Expression
1 imdi 353 . 2  |-  ( ( -.  ph  ->  ( ps 
->  ch ) )  <->  ( ( -.  ph  ->  ps )  ->  ( -.  ph  ->  ch ) ) )
2 df-or 360 . 2  |-  ( (
ph  \/  ( ps  ->  ch ) )  <->  ( -.  ph 
->  ( ps  ->  ch ) ) )
3 df-or 360 . . 3  |-  ( (
ph  \/  ps )  <->  ( -.  ph  ->  ps )
)
4 df-or 360 . . 3  |-  ( (
ph  \/  ch )  <->  ( -.  ph  ->  ch )
)
53, 4imbi12i 317 . 2  |-  ( ( ( ph  \/  ps )  ->  ( ph  \/  ch ) )  <->  ( ( -.  ph  ->  ps )  ->  ( -.  ph  ->  ch ) ) )
61, 2, 53bitr4i 269 1  |-  ( (
ph  \/  ( ps  ->  ch ) )  <->  ( ( ph  \/  ps )  -> 
( ph  \/  ch ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358
This theorem is referenced by:  pm2.76  822  pm2.85  827
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 178  df-or 360
  Copyright terms: Public domain W3C validator