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Theorem untim1d 25123
Description: Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
Hypothesis
Ref Expression
untim1d.1  |-  ( [.] ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
untim1d  |-  ( [.] ph  ->  ( ( ps 
until  th )  ->  ( ch  until  th ) ) )

Proof of Theorem untim1d
StepHypRef Expression
1 untim1d.1 . . . . . 6  |-  ( [.] ph  ->  ( ps  ->  ch ) )
21anim1d 547 . . . . 5  |-  ( [.] ph  ->  ( ( ps 
/\  () ( ch 
until  th ) )  -> 
( ch  /\  () ( ch  until  th )
) ) )
32orim2d 813 . . . 4  |-  ( [.] ph  ->  ( ( th  \/  ( ps  /\  () ( ch  until  th )
) )  ->  ( th  \/  ( ch  /\  () ( ch  until  th )
) ) ) )
4 ax-ltl5 25096 . . . 4  |-  ( ( ch  until  th )  <->  ( th  \/  ( ch  /\  () ( ch  until  th )
) ) )
53, 4syl6ibr 218 . . 3  |-  ( [.] ph  ->  ( ( th  \/  ( ps  /\  () ( ch  until  th )
) )  ->  ( ch  until  th ) ) )
65boxrim 25110 . 2  |-  ( [.] ph  ->  [.] ( ( th  \/  ( ps  /\  () ( ch  until  th )
) )  ->  ( ch  until  th ) ) )
7 untind 25121 . 2  |-  ( [.] ( ( th  \/  ( ps  /\  () ( ch  until  th ) ) )  ->  ( ch  until  th ) )  ->  (
( ps  until  th )  ->  ( ch  until  th )
) )
86, 7syl 15 1  |-  ( [.] ph  ->  ( ( ps 
until  th )  ->  ( ch  until  th ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   [.]wbox 25073   ()wcirc 25075    until wunt 25076
This theorem is referenced by:  untbi12d  25125
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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