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Theorem untind 25121
Description: An "induction principle" for until, roughly stating that it is the least fixed point satisfying a property like ax-ltl5 25096. (Contributed by Mario Carneiro, 30-Aug-2016.)
Assertion
Ref Expression
untind  |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ( ph  until  ps )  ->  th ) )

Proof of Theorem untind
StepHypRef Expression
1 simpr 447 . . 3  |-  ( ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  /\  ( ph  until  ps ) )  ->  ( ph  until  ps )
)
2 ax-ltl5 25096 . . . . . . . 8  |-  ( (
ph  until  ps )  <->  ( ps  \/  ( ph  /\  () ( ph  until  ps ) ) ) )
3 ax-ltl3 25079 . . . . . . . . . . 11  |-  ( () ( ( ph  until  ps )  ->  th )  ->  ( () ( ph  until  ps )  ->  () th ) )
43anim2d 548 . . . . . . . . . 10  |-  ( () ( ( ph  until  ps )  ->  th )  ->  (
( ph  /\  () (
ph  until  ps ) )  ->  ( ph  /\  () th ) ) )
54orim2d 813 . . . . . . . . 9  |-  ( () ( ( ph  until  ps )  ->  th )  ->  (
( ps  \/  ( ph  /\  () ( ph  until  ps ) ) )  -> 
( ps  \/  ( ph  /\  () th )
) ) )
6 alneal1 25103 . . . . . . . . 9  |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ( ps  \/  ( ph  /\  () th ) )  ->  th )
)
75, 6sylan9r 639 . . . . . . . 8  |-  ( ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  /\  () ( ( ph  until  ps )  ->  th ) )  -> 
( ( ps  \/  ( ph  /\  () (
ph  until  ps ) ) )  ->  th )
)
82, 7syl5bi 208 . . . . . . 7  |-  ( ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  /\  () ( ( ph  until  ps )  ->  th ) )  -> 
( ( ph  until  ps )  ->  th ) )
98ex 423 . . . . . 6  |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( () ( (
ph  until  ps )  ->  th )  ->  ( (
ph  until  ps )  ->  th ) ) )
109boxrim 25110 . . . . 5  |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  [.] ( () ( ( ph  until  ps )  ->  th )  ->  (
( ph  until  ps )  ->  th ) ) )
1110adantr 451 . . . 4  |-  ( ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  /\  ( ph  until  ps ) )  ->  [.] ( () ( ( ph  until  ps )  ->  th )  ->  (
( ph  until  ps )  ->  th ) ) )
12 ax-ltl6 25097 . . . . 5  |-  ( (
ph  until  ps )  ->  <> ps )
13 orc 374 . . . . . . . . 9  |-  ( ps 
->  ( ps  \/  ( ph  /\  () th )
) )
1413, 6syl5 28 . . . . . . . 8  |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ps  ->  th )
)
1514a1dd 42 . . . . . . 7  |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ps  ->  (
( ph  until  ps )  ->  th ) ) )
1615diaimd 25113 . . . . . 6  |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( <> ps  ->  <> ( (
ph  until  ps )  ->  th ) ) )
1716imp 418 . . . . 5  |-  ( ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  /\  <> ps )  -> 
<> ( ( ph  until  ps )  ->  th ) )
1812, 17sylan2 460 . . . 4  |-  ( ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  /\  ( ph  until  ps ) )  ->  <> ( ( ph  until  ps )  ->  th )
)
19 ltl4ev 25095 . . . 4  |-  ( ( [.] ( () ( ( ph  until  ps )  ->  th )  ->  (
( ph  until  ps )  ->  th ) )  /\  <> ( ( ph  until  ps )  ->  th ) )  -> 
( ( ph  until  ps )  ->  th ) )
2011, 18, 19syl2anc 642 . . 3  |-  ( ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  /\  ( ph  until  ps ) )  ->  ( ( ph  until  ps )  ->  th )
)
211, 20mpd 14 . 2  |-  ( ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  /\  ( ph  until  ps ) )  ->  th )
2221ex 423 1  |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ( ph  until  ps )  ->  th ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   [.]wbox 25073   <>wdia 25074   ()wcirc 25075    until wunt 25076
This theorem is referenced by:  untindd  25122  untim1d  25123  untim2d  25124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ltl1 25077  ax-ltl2 25078  ax-ltl3 25079  ax-ltl4 25080  ax-lmp 25081  ax-nmp 25082  ax-ltl5 25096  ax-ltl6 25097
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-dia 25083
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