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Theorem nxtimd 25009
Description: Distribute 'next' over implication when the context is always true. (Contributed by Mario Carneiro, 30-Aug-2016.)
Hypothesis
Ref Expression
boximd.1  |-  ( [.] ph  ->  ( ps  ->  ch ) )
Assertion
Ref Expression
nxtimd  |-  ( [.] ph  ->  ( () ps  ->  () ch ) )

Proof of Theorem nxtimd
StepHypRef Expression
1 boximd.1 . . 3  |-  ( [.] ph  ->  ( ps  ->  ch ) )
21boxrim 25007 . 2  |-  ( [.] ph  ->  [.] ( ps  ->  ch ) )
3 alne 25002 . 2  |-  ( [.] ( ps  ->  ch )  ->  () ( ps 
->  ch ) )
4 ax-ltl3 24976 . 2  |-  ( () ( ps  ->  ch )  ->  ( () ps  ->  () ch ) )
52, 3, 43syl 18 1  |-  ( [.] ph  ->  ( () ps  ->  () ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   [.]wbox 24970   ()wcirc 24972
This theorem is referenced by:  nxtbid  25012
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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