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Theorem meran1 24850
Description: A single axiom for propositional calculus offered by Meredith. (Contributed by Anthony Hart, 13-Aug-2011.)
Assertion
Ref Expression
meran1  |-  ( -.  ( -.  ( -. 
ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  th  \/  ph )  \/  ( ch  \/  ( ta  \/  ph ) ) ) )

Proof of Theorem meran1
StepHypRef Expression
1 orc 374 . . . . . 6  |-  ( -. 
ph  ->  ( -.  ph  \/  ps ) )
2 olc 373 . . . . . 6  |-  ( ps 
->  ( -.  ph  \/  ps ) )
31, 2ja 153 . . . . 5  |-  ( (
ph  ->  ps )  -> 
( -.  ph  \/  ps ) )
43imim1i 54 . . . 4  |-  ( ( ( -.  ph  \/  ps )  ->  ( ch  \/  ( th  \/  ta ) ) )  -> 
( ( ph  ->  ps )  ->  ( ch  \/  ( th  \/  ta ) ) ) )
5 pm2.24 101 . . . . . 6  |-  ( th 
->  ( -.  th  ->  ph ) )
6 idd 21 . . . . . 6  |-  ( th 
->  ( ph  ->  ph )
)
75, 6jaod 369 . . . . 5  |-  ( th 
->  ( ( -.  th  \/  ph )  ->  ph )
)
87com12 27 . . . 4  |-  ( ( -.  th  \/  ph )  ->  ( th  ->  ph ) )
9 pm1.5 508 . . . . . 6  |-  ( ( -.  ( ph  ->  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  -> 
( ch  \/  ( -.  ( ph  ->  ps )  \/  ( th  \/  ta ) ) ) )
10 pm2.3 555 . . . . . . . 8  |-  ( ( -.  ( ph  ->  ps )  \/  ( th  \/  ta ) )  ->  ( -.  ( ph  ->  ps )  \/  ( ta  \/  th ) ) )
11 pm1.5 508 . . . . . . . . 9  |-  ( ( -.  ( ph  ->  ps )  \/  ( ta  \/  th ) )  ->  ( ta  \/  ( -.  ( ph  ->  ps )  \/  th ) ) )
12 pm2.21 100 . . . . . . . . . . . . . 14  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
13 mth8 138 . . . . . . . . . . . . . 14  |-  ( th 
->  ( -.  ph  ->  -.  ( th  ->  ph )
) )
1412, 13imim12i 53 . . . . . . . . . . . . 13  |-  ( ( ( ph  ->  ps )  ->  th )  ->  ( -.  ph  ->  ( -.  ph 
->  -.  ( th  ->  ph ) ) ) )
1514pm2.43d 44 . . . . . . . . . . . 12  |-  ( ( ( ph  ->  ps )  ->  th )  ->  ( -.  ph  ->  -.  ( th  ->  ph ) ) )
1615con4d 97 . . . . . . . . . . 11  |-  ( ( ( ph  ->  ps )  ->  th )  ->  (
( th  ->  ph )  ->  ph ) )
17 imor 401 . . . . . . . . . . 11  |-  ( ( ( ph  ->  ps )  ->  th )  <->  ( -.  ( ph  ->  ps )  \/  th ) )
18 imor 401 . . . . . . . . . . 11  |-  ( ( ( th  ->  ph )  ->  ph )  <->  ( -.  ( th  ->  ph )  \/ 
ph ) )
1916, 17, 183imtr3i 256 . . . . . . . . . 10  |-  ( ( -.  ( ph  ->  ps )  \/  th )  ->  ( -.  ( th 
->  ph )  \/  ph ) )
2019orim2i 504 . . . . . . . . 9  |-  ( ( ta  \/  ( -.  ( ph  ->  ps )  \/  th )
)  ->  ( ta  \/  ( -.  ( th 
->  ph )  \/  ph ) ) )
21 pm1.5 508 . . . . . . . . 9  |-  ( ( ta  \/  ( -.  ( th  ->  ph )  \/  ph ) )  -> 
( -.  ( th 
->  ph )  \/  ( ta  \/  ph ) ) )
2211, 20, 213syl 18 . . . . . . . 8  |-  ( ( -.  ( ph  ->  ps )  \/  ( ta  \/  th ) )  ->  ( -.  ( th  ->  ph )  \/  ( ta  \/  ph ) ) )
2310, 22syl 15 . . . . . . 7  |-  ( ( -.  ( ph  ->  ps )  \/  ( th  \/  ta ) )  ->  ( -.  ( th  ->  ph )  \/  ( ta  \/  ph ) ) )
2423orim2i 504 . . . . . 6  |-  ( ( ch  \/  ( -.  ( ph  ->  ps )  \/  ( th  \/  ta ) ) )  ->  ( ch  \/  ( -.  ( th  ->  ph )  \/  ( ta  \/  ph ) ) ) )
25 pm1.5 508 . . . . . 6  |-  ( ( ch  \/  ( -.  ( th  ->  ph )  \/  ( ta  \/  ph ) ) )  -> 
( -.  ( th 
->  ph )  \/  ( ch  \/  ( ta  \/  ph ) ) ) )
269, 24, 253syl 18 . . . . 5  |-  ( ( -.  ( ph  ->  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  -> 
( -.  ( th 
->  ph )  \/  ( ch  \/  ( ta  \/  ph ) ) ) )
27 imor 401 . . . . 5  |-  ( ( ( ph  ->  ps )  ->  ( ch  \/  ( th  \/  ta )
) )  <->  ( -.  ( ph  ->  ps )  \/  ( ch  \/  ( th  \/  ta ) ) ) )
28 imor 401 . . . . 5  |-  ( ( ( th  ->  ph )  ->  ( ch  \/  ( ta  \/  ph ) ) )  <->  ( -.  ( th  ->  ph )  \/  ( ch  \/  ( ta  \/  ph ) ) ) )
2926, 27, 283imtr4i 257 . . . 4  |-  ( ( ( ph  ->  ps )  ->  ( ch  \/  ( th  \/  ta )
) )  ->  (
( th  ->  ph )  ->  ( ch  \/  ( ta  \/  ph ) ) ) )
304, 8, 29syl2im 34 . . 3  |-  ( ( ( -.  ph  \/  ps )  ->  ( ch  \/  ( th  \/  ta ) ) )  -> 
( ( -.  th  \/  ph )  ->  ( ch  \/  ( ta  \/  ph ) ) ) )
31 imor 401 . . 3  |-  ( ( ( -.  ph  \/  ps )  ->  ( ch  \/  ( th  \/  ta ) ) )  <->  ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) ) )
32 imor 401 . . 3  |-  ( ( ( -.  th  \/  ph )  ->  ( ch  \/  ( ta  \/  ph ) ) )  <->  ( -.  ( -.  th  \/  ph )  \/  ( ch  \/  ( ta  \/  ph ) ) ) )
3330, 31, 323imtr3i 256 . 2  |-  ( ( -.  ( -.  ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  -> 
( -.  ( -. 
th  \/  ph )  \/  ( ch  \/  ( ta  \/  ph ) ) ) )
3433imori 402 1  |-  ( -.  ( -.  ( -. 
ph  \/  ps )  \/  ( ch  \/  ( th  \/  ta ) ) )  \/  ( -.  ( -.  th  \/  ph )  \/  ( ch  \/  ( ta  \/  ph ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357
This theorem is referenced by:  meran2  24851  meran3  24852
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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