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

Theorem diaimd 25010
Description: Distribute 'eventually' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
Hypothesis
Ref Expression
boximd.1  |-  ( [.] ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
diaimd  |-  ( [.] ph  ->  ( <> ps  ->  <> ch ) )

Proof of Theorem diaimd
StepHypRef Expression
1 boximd.1 . . . . 5  |-  ( [.] ph  ->  ( ps  ->  ch ) )
21con3d 125 . . . 4  |-  ( [.] ph  ->  ( -.  ch  ->  -.  ps ) )
32boximd 25008 . . 3  |-  ( [.] ph  ->  ( [.]  -.  ch  ->  [.]  -.  ps )
)
43con3d 125 . 2  |-  ( [.] ph  ->  ( -.  [.]  -. 
ps  ->  -.  [.]  -.  ch ) )
5 df-dia 24980 . 2  |-  ( <> ps  <->  -. 
[.]  -.  ps )
6 df-dia 24980 . 2  |-  ( <> ch  <->  -. 
[.]  -.  ch )
74, 5, 63imtr4g 261 1  |-  ( [.] ph  ->  ( <> ps  ->  <> ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   [.]wbox 24970   <>wdia 24971
This theorem is referenced by:  diabid  25013  untind  25018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ltl1 24974  ax-ltl2 24975  ax-ltl3 24976  ax-ltl4 24977  ax-lmp 24978  ax-nmp 24979  ax-ltl5 24993  ax-ltl6 24994
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-dia 24980
  Copyright terms: Public domain W3C validator