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Theorem albineal 24999
Description:  ph always holds iff  ph holds in the first step and always holds in the next step. (Contributed by FL, 20-Mar-2011.)
Assertion
Ref Expression
albineal  |-  ( [.] ph 
<->  ( ph  /\  () [.] ph ) )

Proof of Theorem albineal
StepHypRef Expression
1 boxeq 24987 . 2  |-  ( [.] ph 
<->  -.  <>  -.  ph )
2 evpexun 24998 . . . 4  |-  ( <>  -.  ph  <->  (  T.  until  -.  ph ) )
3 ax-ltl5 24993 . . . 4  |-  ( (  T.  until  -.  ph )  <->  ( -.  ph  \/  (  T.  /\  () (  T.  until  -.  ph ) ) ) )
42, 3bitri 240 . . 3  |-  ( <>  -.  ph  <->  ( -.  ph  \/  (  T.  /\  () (  T. 
until  -.  ph ) ) ) )
54notbii 287 . 2  |-  ( -.  <> 
-.  ph  <->  -.  ( -.  ph  \/  (  T.  /\  () (  T.  until  -.  ph ) ) ) )
6 ioran 476 . . 3  |-  ( -.  ( -.  ph  \/  (  T.  /\  () (  T.  until  -.  ph ) ) )  <->  ( -.  -.  ph 
/\  -.  (  T.  /\  () (  T.  until  -. 
ph ) ) ) )
7 notnot 282 . . . . 5  |-  ( ph  <->  -. 
-.  ph )
87bicomi 193 . . . 4  |-  ( -. 
-.  ph  <->  ph )
9 trcrm 24951 . . . . . 6  |-  ( (  T.  /\  () (  T.  until  -.  ph ) )  <-> 
() (  T.  until  -. 
ph ) )
109notbii 287 . . . . 5  |-  ( -.  (  T.  /\  () (  T.  until  -.  ph ) )  <->  -.  () (  T.  until  -.  ph ) )
11 ax-ltl2 24975 . . . . 5  |-  ( -.  () (  T.  until  -. 
ph )  <->  ()  -.  (  T.  until  -.  ph ) )
122notbii 287 . . . . . . 7  |-  ( -.  <> 
-.  ph  <->  -.  (  T.  until  -.  ph ) )
131, 12bitr2i 241 . . . . . 6  |-  ( -.  (  T.  until  -.  ph ) 
<->  [.] ph )
1413binxt 24984 . . . . 5  |-  ( () 
-.  (  T.  until  -. 
ph )  <->  () [.] ph )
1510, 11, 143bitri 262 . . . 4  |-  ( -.  (  T.  /\  () (  T.  until  -.  ph ) )  <->  () [.] ph )
168, 15anbi12i 678 . . 3  |-  ( ( -.  -.  ph  /\  -.  (  T.  /\  () (  T.  until  -.  ph ) ) )  <->  ( ph  /\  () [.] ph )
)
176, 16bitri 240 . 2  |-  ( -.  ( -.  ph  \/  (  T.  /\  () (  T.  until  -.  ph ) ) )  <->  ( ph  /\  () [.] ph ) )
181, 5, 173bitri 262 1  |-  ( [.] ph 
<->  ( ph  /\  () [.] ph ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    T. wtru 1307   [.]wbox 24970   <>wdia 24971   ()wcirc 24972    until wunt 24973
This theorem is referenced by:  alneal1  25000  alneal2  25001
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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