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

Theorem jaob 758
Description: Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
Assertion
Ref Expression
jaob  |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
) )

Proof of Theorem jaob
StepHypRef Expression
1 pm2.67-2 391 . . 3  |-  ( ( ( ph  \/  ch )  ->  ps )  -> 
( ph  ->  ps )
)
2 olc 373 . . . 4  |-  ( ch 
->  ( ph  \/  ch ) )
32imim1i 54 . . 3  |-  ( ( ( ph  \/  ch )  ->  ps )  -> 
( ch  ->  ps ) )
41, 3jca 518 . 2  |-  ( ( ( ph  \/  ch )  ->  ps )  -> 
( ( ph  ->  ps )  /\  ( ch 
->  ps ) ) )
5 pm3.44 497 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ch  ->  ps ) )  ->  (
( ph  \/  ch )  ->  ps ) )
64, 5impbii 180 1  |-  ( ( ( ph  \/  ch )  ->  ps )  <->  ( ( ph  ->  ps )  /\  ( ch  ->  ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358
This theorem is referenced by:  pm4.77  762  pm5.53  771  pm4.83  895  unss  3349  ralunb  3356  intun  3894  intpr  3895  relop  4834  sqr2irr  12527  algcvgblem  12747  efgred  15057  caucfil  18709  plydivex  19677  2sqlem6  20608  arg-ax  24855  domrngref  25060  a12study4  29117  tendoeq2  30963
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  df-an 360
  Copyright terms: Public domain W3C validator