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Theorem untindd 25122
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.)
Hypotheses
Ref Expression
untindd.1  |-  ( ps 
->  th )
untindd.2  |-  ( (
ph  /\  () th )  ->  th )
Assertion
Ref Expression
untindd  |-  ( (
ph  until  ps )  ->  th )

Proof of Theorem untindd
StepHypRef Expression
1 untindd.1 . . . 4  |-  ( ps 
->  th )
2 untindd.2 . . . 4  |-  ( (
ph  /\  () th )  ->  th )
31, 2jaoi 368 . . 3  |-  ( ( ps  \/  ( ph  /\  () th ) )  ->  th )
43ax-lmp 25081 . 2  |-  [.] (
( ps  \/  ( ph  /\  () th )
)  ->  th )
5 untind 25121 . 2  |-  ( [.] ( ( ps  \/  ( ph  /\  () th ) )  ->  th )  ->  ( ( ph  until  ps )  ->  th ) )
64, 5ax-mp 8 1  |-  ( (
ph  until  ps )  ->  th )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   [.]wbox 25073   ()wcirc 25075    until wunt 25076
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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