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Theorem untbi12i 25126
Description: Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
Hypotheses
Ref Expression
untbi12i.1  |-  ( ph  <->  ps )
untbi12i.2  |-  ( ch  <->  th )
Assertion
Ref Expression
untbi12i  |-  ( (
ph  until  ch )  <->  ( ps  until  th ) )

Proof of Theorem untbi12i
StepHypRef Expression
1 trtrst 25119 . 2  |-  [.]  T.
2 untbi12i.1 . . . 4  |-  ( ph  <->  ps )
32a1i 10 . . 3  |-  ( [.] 
T.  ->  ( ph  <->  ps )
)
4 untbi12i.2 . . . 4  |-  ( ch  <->  th )
54a1i 10 . . 3  |-  ( [.] 
T.  ->  ( ch  <->  th )
)
63, 5untbi12d 25125 . 2  |-  ( [.] 
T.  ->  ( ( ph  until  ch )  <->  ( ps  until  th ) ) )
71, 6ax-mp 8 1  |-  ( (
ph  until  ch )  <->  ( ps  until  th ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    T. wtru 1307   [.]wbox 25073    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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