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Theorem boxeq 24987
Description:  ph holds now and will always hold in the future iff it is not true that  -.  ph holds now or sometimes in the future. (Contributed by FL, 22-Feb-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
boxeq  |-  ( [.] ph 
<->  -.  <>  -.  ph )

Proof of Theorem boxeq
StepHypRef Expression
1 df-dia 24980 . . 3  |-  ( <>  -.  ph  <->  -. 
[.]  -.  -.  ph )
2 notnot 282 . . . . 5  |-  ( ph  <->  -. 
-.  ph )
32bicomi 193 . . . 4  |-  ( -. 
-.  ph  <->  ph )
43bibox 24982 . . 3  |-  ( [.] 
-.  -.  ph  <->  [.] ph )
51, 4xchbinx 301 . 2  |-  ( <>  -.  ph  <->  -. 
[.] ph )
65con2bii 322 1  |-  ( [.] ph 
<->  -.  <>  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   [.]wbox 24970   <>wdia 24971
This theorem is referenced by:  notal  24991  albineal  24999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ltl1 24974  ax-lmp 24978
This theorem depends on definitions:  df-bi 177  df-dia 24980
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