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Theorem untbi12d 25125
Description: Congruence axiom for until. (Contributed by Mario Carneiro, 30-Aug-2016.)
Hypotheses
Ref Expression
untbi12d.1  |-  ( [.] ph  ->  ( ps  <->  ch )
)
untbi12d.2  |-  ( [.] ph  ->  ( th  <->  ta )
)
Assertion
Ref Expression
untbi12d  |-  ( [.] ph  ->  ( ( ps 
until  th )  <->  ( ch  until  ta ) ) )

Proof of Theorem untbi12d
StepHypRef Expression
1 untbi12d.1 . . . . 5  |-  ( [.] ph  ->  ( ps  <->  ch )
)
21biimpd 198 . . . 4  |-  ( [.] ph  ->  ( ps  ->  ch ) )
32untim1d 25123 . . 3  |-  ( [.] ph  ->  ( ( ps 
until  th )  ->  ( ch  until  th ) ) )
41biimprd 214 . . . 4  |-  ( [.] ph  ->  ( ch  ->  ps ) )
54untim1d 25123 . . 3  |-  ( [.] ph  ->  ( ( ch 
until  th )  ->  ( ps  until  th ) ) )
63, 5impbid 183 . 2  |-  ( [.] ph  ->  ( ( ps 
until  th )  <->  ( ch  until  th ) ) )
7 untbi12d.2 . . . . 5  |-  ( [.] ph  ->  ( th  <->  ta )
)
87biimpd 198 . . . 4  |-  ( [.] ph  ->  ( th  ->  ta ) )
98untim2d 25124 . . 3  |-  ( [.] ph  ->  ( ( ch 
until  th )  ->  ( ch  until  ta ) ) )
107biimprd 214 . . . 4  |-  ( [.] ph  ->  ( ta  ->  th ) )
1110untim2d 25124 . . 3  |-  ( [.] ph  ->  ( ( ch 
until  ta )  ->  ( ch  until  th ) ) )
129, 11impbid 183 . 2  |-  ( [.] ph  ->  ( ( ch 
until  th )  <->  ( ch  until  ta ) ) )
136, 12bitrd 244 1  |-  ( [.] ph  ->  ( ( ps 
until  th )  <->  ( ch  until  ta ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   [.]wbox 25073    until wunt 25076
This theorem is referenced by:  untbi12i  25126  cdequnt  25134
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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