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Theorem boxrim 25007
Description: If  [.] ph implies  ps in the current world, then it implies  ps in every world. (Contributed by Mario Carneiro, 30-Aug-2016.)
Hypothesis
Ref Expression
boxrim.1  |-  ( [.] ph  ->  ps )
Assertion
Ref Expression
boxrim  |-  ( [.] ph  ->  [.] ps )

Proof of Theorem boxrim
StepHypRef Expression
1 alalifal 25003 . 2  |-  ( [.]
[.] ph  <->  [.] ph )
2 boxrim.1 . . 3  |-  ( [.] ph  ->  ps )
32impbox 24981 . 2  |-  ( [.]
[.] ph  ->  [.] ps )
41, 3sylbir 204 1  |-  ( [.] ph  ->  [.] ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   [.]wbox 24970
This theorem is referenced by:  boximd  25008  nxtimd  25009  althalne  25015  untind  25018  untim1d  25020  untim2d  25021
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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