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Theorem untim2d 25021
Description: Congruence axiom for until. (Contributed by Mario Carneiro, 11-Aug-2016.)
Hypothesis
Ref Expression
untim1d.1  |-  ( [.] ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
untim2d  |-  ( [.] ph  ->  ( ( th 
until  ps )  ->  ( th  until  ch ) ) )

Proof of Theorem untim2d
StepHypRef Expression
1 untim1d.1 . . . . 5  |-  ( [.] ph  ->  ( ps  ->  ch ) )
21orim1d 812 . . . 4  |-  ( [.] ph  ->  ( ( ps  \/  ( th  /\  () ( th  until  ch )
) )  ->  ( ch  \/  ( th  /\  () ( th  until  ch )
) ) ) )
3 ax-ltl5 24993 . . . 4  |-  ( ( th  until  ch )  <->  ( ch  \/  ( th  /\  () ( th  until  ch ) ) ) )
42, 3syl6ibr 218 . . 3  |-  ( [.] ph  ->  ( ( ps  \/  ( th  /\  () ( th  until  ch )
) )  ->  ( th  until  ch ) ) )
54boxrim 25007 . 2  |-  ( [.] ph  ->  [.] ( ( ps  \/  ( th  /\  () ( th  until  ch )
) )  ->  ( th  until  ch ) ) )
6 untind 25018 . 2  |-  ( [.] ( ( ps  \/  ( th  /\  () ( th  until  ch ) ) )  ->  ( th  until  ch ) )  ->  (
( th  until  ps )  ->  ( th  until  ch )
) )
75, 6syl 15 1  |-  ( [.] ph  ->  ( ( th 
until  ps )  ->  ( th  until  ch ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    /\ wa 358   [.]wbox 24970   ()wcirc 24972    until wunt 24973
This theorem is referenced by:  untbi12d  25022
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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