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

Theorem imdistan 671
Description: Distribution of implication with conjunction. (Contributed by NM, 31-May-1999.) (Proof shortened by Wolf Lammen, 6-Dec-2012.)
Assertion
Ref Expression
imdistan  |-  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( ( ph  /\  ps )  -> 
( ph  /\  ch )
) )

Proof of Theorem imdistan
StepHypRef Expression
1 pm5.42 532 . 2  |-  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( ph  ->  ( ps  ->  ( ph  /\  ch ) ) ) )
2 impexp 434 . 2  |-  ( ( ( ph  /\  ps )  ->  ( ph  /\  ch ) )  <->  ( ph  ->  ( ps  ->  ( ph  /\  ch ) ) ) )
31, 2bitr4i 244 1  |-  ( (
ph  ->  ( ps  ->  ch ) )  <->  ( ( ph  /\  ps )  -> 
( ph  /\  ch )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359
This theorem is referenced by:  imdistand  674  pm5.3  693  rmoim  3133  ss2rab  3419  marypha2lem3  7442
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-an 361
  Copyright terms: Public domain W3C validator