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Theorem alalifal 25003
Description: It is always true that  ph always holds iff  ph always holds. (Contributed by FL, 20-Mar-2011.)
Assertion
Ref Expression
alalifal  |-  ( [.]
[.] ph  <->  [.] ph )

Proof of Theorem alalifal
StepHypRef Expression
1 alneal1 25000 . 2  |-  ( [.]
[.] ph  ->  [.] ph )
2 alneal2 25001 . . . . 5  |-  ( [.] ph  ->  () [.] ph )
32a1i 10 . . . 4  |-  ( ph  ->  ( [.] ph  ->  ()
[.] ph ) )
43impbox 24981 . . 3  |-  ( [.] ph  ->  [.] ( [.] ph  ->  () [.] ph )
)
5 ax-ltl4 24977 . . 3  |-  ( ( [.] ( [.] ph  ->  () [.] ph )  /\  [.] ph )  ->  [.] [.] ph )
64, 5mpancom 650 . 2  |-  ( [.] ph  ->  [.] [.] ph )
71, 6impbii 180 1  |-  ( [.]
[.] ph  <->  [.] ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   [.]wbox 24970   ()wcirc 24972
This theorem is referenced by:  alneal1a  25004  boxrim  25007  evevifev  25014
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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