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Theorem meran3 24852
Description: A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
Assertion
Ref Expression
meran3  |-  ( -.  ( -.  ( -. 
ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  ch  \/  ph )  \/  ( ta  \/  ( th  \/  ph ) ) ) )

Proof of Theorem meran3
StepHypRef Expression
1 pm2.3 555 . . . . . 6  |-  ( ( ch  \/  ( th  \/  ta ) )  ->  ( ch  \/  ( ta  \/  th )
) )
21imim2i 13 . . . . 5  |-  ( ( ( -.  ph  \/  ps )  ->  ( ch  \/  ( th  \/  ta ) ) )  -> 
( ( -.  ph  \/  ps )  ->  ( ch  \/  ( ta  \/  th ) ) ) )
3 pm1.5 508 . . . . 5  |-  ( ( ch  \/  ( ta  \/  th ) )  ->  ( ta  \/  ( ch  \/  th )
) )
42, 3syl6 29 . . . 4  |-  ( ( ( -.  ph  \/  ps )  ->  ( ch  \/  ( th  \/  ta ) ) )  -> 
( ( -.  ph  \/  ps )  ->  ( ta  \/  ( ch  \/  th ) ) ) )
5 imor 401 . . . 4  |-  ( ( ( -.  ph  \/  ps )  ->  ( ch  \/  ( th  \/  ta ) ) )  <->  ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) ) )
6 imor 401 . . . 4  |-  ( ( ( -.  ph  \/  ps )  ->  ( ta  \/  ( ch  \/  th ) ) )  <->  ( -.  ( -.  ph  \/  ps )  \/  ( ta  \/  ( ch  \/  th ) ) ) )
74, 5, 63imtr3i 256 . . 3  |-  ( ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  -> 
( -.  ( -. 
ph  \/  ps )  \/  ( ta  \/  ( ch  \/  th ) ) ) )
8 meran1 24850 . . . 4  |-  ( -.  ( -.  ( -. 
ph  \/  ps )  \/  ( ta  \/  ( ch  \/  th ) ) )  \/  ( -.  ( -.  ch  \/  ph )  \/  ( ta  \/  ( th  \/  ph ) ) ) )
98imorri 403 . . 3  |-  ( ( -.  ( -.  ph  \/  ps )  \/  ( ta  \/  ( ch  \/  th ) ) )  -> 
( -.  ( -. 
ch  \/  ph )  \/  ( ta  \/  ( th  \/  ph ) ) ) )
107, 9syl 15 . 2  |-  ( ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  -> 
( -.  ( -. 
ch  \/  ph )  \/  ( ta  \/  ( th  \/  ph ) ) ) )
1110imori 402 1  |-  ( -.  ( -.  ( -. 
ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  ch  \/  ph )  \/  ( ta  \/  ( th  \/  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8
This theorem depends on definitions:  df-bi 177  df-or 359
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