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

Theorem impbidd 181
Description: Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
Hypotheses
Ref Expression
impbidd.1  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
impbidd.2  |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
Assertion
Ref Expression
impbidd  |-  ( ph  ->  ( ps  ->  ( ch 
<->  th ) ) )

Proof of Theorem impbidd
StepHypRef Expression
1 impbidd.1 . 2  |-  ( ph  ->  ( ps  ->  ( ch  ->  th ) ) )
2 impbidd.2 . 2  |-  ( ph  ->  ( ps  ->  ( th  ->  ch ) ) )
3 bi3 179 . 2  |-  ( ( ch  ->  th )  ->  ( ( th  ->  ch )  ->  ( ch  <->  th ) ) )
41, 2, 3syl6c 60 1  |-  ( ph  ->  ( ps  ->  ( ch 
<->  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176
This theorem is referenced by:  impbid21d  182  pm5.74  235  seglecgr12  24805  wl-bitr1  24981  impbiddOLD  26806  prtlem18  26847
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
  Copyright terms: Public domain W3C validator