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Theorem oal2 999
Description: Orthoarguesian law - ->2 version.
Assertion
Ref Expression
oal2 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
Proof of Theorem oal2
StepHypRef Expression
1 ax-3oa 998 . 2 ((b' ->1 a') ^ ((b' ^ c') v ((b' ->1 a') ^ (c' ->1 a')))) =< (c' ->1 a')
2 i2i1 267 . . 3 (a ->2 b) = (b' ->1 a')
3 anor3 90 . . . . 5 (b' ^ c') = (b v c)'
43ax-r1 35 . . . 4 (b v c)' = (b' ^ c')
5 i2i1 267 . . . . 5 (a ->2 c) = (c' ->1 a')
62, 52an 79 . . . 4 ((a ->2 b) ^ (a ->2 c)) = ((b' ->1 a') ^ (c' ->1 a'))
74, 62or 72 . . 3 ((b v c)' v ((a ->2 b) ^ (a ->2 c))) = ((b' ^ c') v ((b' ->1 a') ^ (c' ->1 a')))
82, 72an 79 . 2 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((b' ->1 a') ^ ((b' ^ c') v ((b' ->1 a') ^ (c' ->1 a'))))
91, 8, 5le3tr1 140 1 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) =< (a ->2 c)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-3oa 998
This theorem depends on definitions:  df-a 40  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by:  oal1  1000  oaliii  1001  oagen2  1016  mloa  1018  oadistc0  1021
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