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Theorem evevifev 25014
Description: It is eventually true that  ph eventually holds iff  ph eventually holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
evevifev  |-  ( <> <> ph  <->  <> ph )

Proof of Theorem evevifev
StepHypRef Expression
1 notev 24990 . . . . 5  |-  ( -.  <> ph 
<->  [.]  -.  ph )
21bibox 24982 . . . 4  |-  ( [.] 
-.  <> ph  <->  [.] [.]  -.  ph )
3 alalifal 25003 . . . 4  |-  ( [.]
[.]  -.  ph  <->  [.]  -.  ph )
42, 3bitri 240 . . 3  |-  ( [.] 
-.  <> ph  <->  [.]  -.  ph )
54notbii 287 . 2  |-  ( -. 
[.]  -.  <> ph  <->  -.  [.]  -.  ph )
6 df-dia 24980 . 2  |-  ( <> <> ph  <->  -. 
[.]  -.  <> ph )
7 df-dia 24980 . 2  |-  ( <> ph  <->  -. 
[.]  -.  ph )
85, 6, 73bitr4i 268 1  |-  ( <> <> ph  <->  <> ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   [.]wbox 24970   <>wdia 24971
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
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