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Theorem oalii 1002
Description: Orthoarguesian law. Godowski/Greechie, Eq. II. This proof references oaliii 1001 only.
Assertion
Ref Expression
oalii (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< a'
Proof of Theorem oalii
StepHypRef Expression
1 orabs 120 . . . . 5 ((a ->2 b) v ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))) = (a ->2 b)
2 oaliii 1001 . . . . . 6 ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))) = ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))
32lor 70 . . . . 5 ((a ->2 b) v ((a ->2 b) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))) = ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))
4 df-i2 45 . . . . . 6 (a ->2 b) = (b v (a' ^ b'))
5 ancom 74 . . . . . . 7 (a' ^ b') = (b' ^ a')
65lor 70 . . . . . 6 (b v (a' ^ b')) = (b v (b' ^ a'))
74, 6ax-r2 36 . . . . 5 (a ->2 b) = (b v (b' ^ a'))
81, 3, 73tr2 64 . . . 4 ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c))))) = (b v (b' ^ a'))
98lan 77 . . 3 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) = (b' ^ (b v (b' ^ a')))
10 omlan 448 . . 3 (b' ^ (b v (b' ^ a'))) = (b' ^ a')
119, 10ax-r2 36 . 2 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) = (b' ^ a')
12 lear 161 . 2 (b' ^ a') =< a'
1311, 12bltr 138 1 (b' ^ ((a ->2 b) v ((a ->2 c) ^ ((b v c)' v ((a ->2 b) ^ (a ->2 c)))))) =< a'
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->2 wi2 13
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439  ax-3oa 998
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by:  oaliv  1003  oalem1  1005
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