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Theorem nom51 332
Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom51 ((a v b) ==1 b) = (a ->2 b)
Proof of Theorem nom51
StepHypRef Expression
1 ancom 74 . . . . . . . . 9 (b' ^ a') = (a' ^ b')
2 anor3 90 . . . . . . . . 9 (a' ^ b') = (a v b)'
31, 2ax-r2 36 . . . . . . . 8 (b' ^ a') = (a v b)'
43ax-r1 35 . . . . . . 7 (a v b)' = (b' ^ a')
54ax-r4 37 . . . . . 6 (a v b)'' = (b' ^ a')'
65lor 70 . . . . 5 (b' v (a v b)'') = (b' v (b' ^ a')')
72ax-r1 35 . . . . . . . . 9 (a v b)' = (a' ^ b')
8 ancom 74 . . . . . . . . 9 (a' ^ b') = (b' ^ a')
97, 8ax-r2 36 . . . . . . . 8 (a v b)' = (b' ^ a')
109ax-r4 37 . . . . . . 7 (a v b)'' = (b' ^ a')'
1110lan 77 . . . . . 6 (b'' ^ (a v b)'') = (b'' ^ (b' ^ a')')
124, 112or 72 . . . . 5 ((a v b)' v (b'' ^ (a v b)'')) = ((b' ^ a') v (b'' ^ (b' ^ a')'))
136, 122an 79 . . . 4 ((b' v (a v b)'') ^ ((a v b)' v (b'' ^ (a v b)''))) = ((b' v (b' ^ a')') ^ ((b' ^ a') v (b'' ^ (b' ^ a')')))
14 df-id2 51 . . . 4 (b' ==2 (a v b)') = ((b' v (a v b)'') ^ ((a v b)' v (b'' ^ (a v b)'')))
15 df-id2 51 . . . 4 (b' ==2 (b' ^ a')) = ((b' v (b' ^ a')') ^ ((b' ^ a') v (b'' ^ (b' ^ a')')))
1613, 14, 153tr1 63 . . 3 (b' ==2 (a v b)') = (b' ==2 (b' ^ a'))
17 nom22 315 . . 3 (b' ==2 (b' ^ a')) = (b' ->1 a')
1816, 17ax-r2 36 . 2 (b' ==2 (a v b)') = (b' ->1 a')
19 nomcon1 302 . 2 ((a v b) ==1 b) = (b' ==2 (a v b)')
20 i2i1 267 . 2 (a ->2 b) = (b' ->1 a')
2118, 19, 203tr1 63 1 ((a v b) ==1 b) = (a ->2 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->2 wi2 13   ==1 wid1 18   ==2 wid2 19
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
This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-id1 50  df-id2 51  df-le1 130  df-le2 131
This theorem is referenced by:  nom64  341
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