MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pm4.79 Structured version   Unicode version

Theorem pm4.79 568
Description: Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)
Assertion
Ref Expression
pm4.79  |-  ( ( ( ps  ->  ph )  \/  ( ch  ->  ph )
)  <->  ( ( ps 
/\  ch )  ->  ph )
)

Proof of Theorem pm4.79
StepHypRef Expression
1 id 21 . . 3  |-  ( ( ps  ->  ph )  -> 
( ps  ->  ph )
)
2 id 21 . . 3  |-  ( ( ch  ->  ph )  -> 
( ch  ->  ph )
)
31, 2jaoa 498 . 2  |-  ( ( ( ps  ->  ph )  \/  ( ch  ->  ph )
)  ->  ( ( ps  /\  ch )  ->  ph ) )
4 simplim 146 . . . 4  |-  ( -.  ( ps  ->  ph )  ->  ps )
5 pm3.3 433 . . . 4  |-  ( ( ( ps  /\  ch )  ->  ph )  ->  ( ps  ->  ( ch  ->  ph ) ) )
64, 5syl5 31 . . 3  |-  ( ( ( ps  /\  ch )  ->  ph )  ->  ( -.  ( ps  ->  ph )  ->  ( ch  ->  ph )
) )
76orrd 369 . 2  |-  ( ( ( ps  /\  ch )  ->  ph )  ->  (
( ps  ->  ph )  \/  ( ch  ->  ph )
) )
83, 7impbii 182 1  |-  ( ( ( ps  ->  ph )  \/  ( ch  ->  ph )
)  <->  ( ( ps 
/\  ch )  ->  ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360
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 179  df-or 361  df-an 362
  Copyright terms: Public domain W3C validator