Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bidia Unicode version

Theorem bidia 24989
Description: If  ph  <->  ps holds then  ph eventually holds iff  ps eventually holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
Hypothesis
Ref Expression
bidia.1  |-  ( ph  <->  ps )
Assertion
Ref Expression
bidia  |-  ( <> ph  <->  <> ps )

Proof of Theorem bidia
StepHypRef Expression
1 bidia.1 . . . 4  |-  ( ph  <->  ps )
21biimpi 186 . . 3  |-  ( ph  ->  ps )
32diaimi 24988 . 2  |-  ( <> ph  -> 
<> ps )
41biimpri 197 . . 3  |-  ( ps 
->  ph )
54diaimi 24988 . 2  |-  ( <> ps  -> 
<> ph )
63, 5impbii 180 1  |-  ( <> ph  <->  <> ps )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   <>wdia 24971
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ltl1 24974  ax-lmp 24978
This theorem depends on definitions:  df-bi 177  df-dia 24980
  Copyright terms: Public domain W3C validator