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Theorem nopsthph 25098
Description: If  ps doesn't hold in the first step and  ph holds until  ps then  ph holds. (Contributed by FL, 20-Mar-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
nopsthph  |-  ( ( -.  ps  /\  ( ph  until  ps ) )  ->  ph )

Proof of Theorem nopsthph
StepHypRef Expression
1 ax-ltl5 25096 . . 3  |-  ( (
ph  until  ps )  <->  ( ps  \/  ( ph  /\  () ( ph  until  ps ) ) ) )
2 simpl 443 . . . . 5  |-  ( (
ph  /\  () ( ph  until  ps ) )  ->  ph )
32orim2i 504 . . . 4  |-  ( ( ps  \/  ( ph  /\  () ( ph  until  ps )
) )  ->  ( ps  \/  ph ) )
43ord 366 . . 3  |-  ( ( ps  \/  ( ph  /\  () ( ph  until  ps )
) )  ->  ( -.  ps  ->  ph ) )
51, 4sylbi 187 . 2  |-  ( (
ph  until  ps )  -> 
( -.  ps  ->  ph ) )
65impcom 419 1  |-  ( ( -.  ps  /\  ( ph  until  ps ) )  ->  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358   ()wcirc 25075    until wunt 25076
This theorem is referenced by:  phthps  25099
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-ltl5 25096
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360
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