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Theorem boxbid 25011
Description: Distribute 'always' over biconditional when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
Hypothesis
Ref Expression
boxbid.1  |-  ( [.] ph  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
boxbid  |-  ( [.] ph  ->  ( [.] ps  <->  [.]
ch ) )

Proof of Theorem boxbid
StepHypRef Expression
1 boxbid.1 . . . 4  |-  ( [.] ph  ->  ( ps  <->  ch )
)
21biimpd 198 . . 3  |-  ( [.] ph  ->  ( ps  ->  ch ) )
32boximd 25008 . 2  |-  ( [.] ph  ->  ( [.] ps  ->  [.] ch ) )
41biimprd 214 . . 3  |-  ( [.] ph  ->  ( ch  ->  ps ) )
54boximd 25008 . 2  |-  ( [.] ph  ->  ( [.] ch  ->  [.] ps ) )
63, 5impbid 183 1  |-  ( [.] ph  ->  ( [.] ps  <->  [.]
ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   [.]wbox 24970
This theorem is referenced by:  cdeqbox  25029
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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