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Theorem diaimi 25091
Description: If  ph implies  ps unconditionally, then if  ph eventually holds so does  ps. (Contributed by Mario Carneiro, 30-Aug-2016.)
Hypothesis
Ref Expression
diaimi.1  |-  ( ph  ->  ps )
Assertion
Ref Expression
diaimi  |-  ( <> ph  -> 
<> ps )

Proof of Theorem diaimi
StepHypRef Expression
1 diaimi.1 . . . . 5  |-  ( ph  ->  ps )
21con3i 127 . . . 4  |-  ( -. 
ps  ->  -.  ph )
32impbox 25084 . . 3  |-  ( [.] 
-.  ps  ->  [.]  -.  ph )
43con3i 127 . 2  |-  ( -. 
[.]  -.  ph  ->  -.  [.]  -.  ps )
5 df-dia 25083 . 2  |-  ( <> ph  <->  -. 
[.]  -.  ph )
6 df-dia 25083 . 2  |-  ( <> ps  <->  -. 
[.]  -.  ps )
74, 5, 63imtr4i 257 1  |-  ( <> ph  -> 
<> ps )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   [.]wbox 25073   <>wdia 25074
This theorem is referenced by:  bidia  25092
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ltl1 25077  ax-lmp 25081
This theorem depends on definitions:  df-bi 177  df-dia 25083
  Copyright terms: Public domain W3C validator