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Theorem evpexun 24998
Description: Eventually  ph expressed with the  until operator. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Mario Carneiro, 30-Aug-2016.)
Assertion
Ref Expression
evpexun  |-  ( <> ph  <->  (  T.  until  ph ) )

Proof of Theorem evpexun
StepHypRef Expression
1 trcrm 24951 . . . . . . 7  |-  ( (  T.  /\  () (  T.  until  ph ) )  <->  () (  T.  until  ph ) )
21biimpri 197 . . . . . 6  |-  ( () (  T.  until  ph )  ->  (  T.  /\  () (  T.  until  ph )
) )
32olcd 382 . . . . 5  |-  ( () (  T.  until  ph )  ->  ( ph  \/  (  T.  /\  () (  T. 
until  ph ) ) ) )
4 ax-ltl5 24993 . . . . 5  |-  ( (  T.  until  ph )  <->  ( ph  \/  (  T.  /\  () (  T.  until  ph )
) ) )
53, 4sylibr 203 . . . 4  |-  ( () (  T.  until  ph )  ->  (  T.  until  ph )
)
65ax-lmp 24978 . . 3  |-  [.] ( () (  T.  until  ph )  ->  (  T.  until  ph )
)
7 orc 374 . . . . . . . 8  |-  ( ph  ->  ( ph  \/  (  T.  /\  () (  T. 
until  ph ) ) ) )
87, 4sylibr 203 . . . . . . 7  |-  ( ph  ->  (  T.  until  ph )
)
98con3i 127 . . . . . 6  |-  ( -.  (  T.  until  ph )  ->  -.  ph )
109impbox 24981 . . . . 5  |-  ( [.] 
-.  (  T.  until  ph )  ->  [.]  -.  ph )
11 notev 24990 . . . . 5  |-  ( -.  <> (  T.  until  ph )  <->  [.] 
-.  (  T.  until  ph ) )
12 notev 24990 . . . . 5  |-  ( -.  <> ph 
<->  [.]  -.  ph )
1310, 11, 123imtr4i 257 . . . 4  |-  ( -.  <> (  T.  until  ph )  ->  -.  <> ph )
1413con4i 122 . . 3  |-  ( <> ph  -> 
<> (  T.  until  ph )
)
15 ltl4ev 24992 . . 3  |-  ( ( [.] ( () (  T.  until  ph )  ->  (  T.  until  ph ) )  /\  <> (  T.  until  ph )
)  ->  (  T.  until  ph ) )
166, 14, 15sylancr 644 . 2  |-  ( <> ph  ->  (  T.  until  ph )
)
17 ax-ltl6 24994 . 2  |-  ( (  T.  until  ph )  ->  <> ph )
1816, 17impbii 180 1  |-  ( <> ph  <->  (  T.  until  ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    T. wtru 1307   [.]wbox 24970   <>wdia 24971   ()wcirc 24972    until wunt 24973
This theorem is referenced by:  albineal  24999
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-ltl4 24977  ax-lmp 24978  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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