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Theorem nom53 334
Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom53 ((a v b) ==3 b) = (a ->2 b)
Proof of Theorem nom53
StepHypRef Expression
1 ancom 74 . . . . . . . 8 (b' ^ a') = (a' ^ b')
2 anor3 90 . . . . . . . 8 (a' ^ b') = (a v b)'
31, 2ax-r2 36 . . . . . . 7 (b' ^ a') = (a v b)'
43ax-r1 35 . . . . . 6 (a v b)' = (b' ^ a')
54lor 70 . . . . 5 (b'' v (a v b)') = (b'' v (b' ^ a'))
64ax-r4 37 . . . . . 6 (a v b)'' = (b' ^ a')'
74lan 77 . . . . . 6 (b' ^ (a v b)') = (b' ^ (b' ^ a'))
86, 72or 72 . . . . 5 ((a v b)'' v (b' ^ (a v b)')) = ((b' ^ a')' v (b' ^ (b' ^ a')))
95, 82an 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'))))
10 df-id4 53 . . . 4 (b' ==4 (a v b)') = ((b'' v (a v b)') ^ ((a v b)'' v (b' ^ (a v b)')))
11 df-id4 53 . . . 4 (b' ==4 (b' ^ a')) = ((b'' v (b' ^ a')) ^ ((b' ^ a')' v (b' ^ (b' ^ a'))))
129, 10, 113tr1 63 . . 3 (b' ==4 (a v b)') = (b' ==4 (b' ^ a'))
13 nom24 317 . . 3 (b' ==4 (b' ^ a')) = (b' ->1 a')
1412, 13ax-r2 36 . 2 (b' ==4 (a v b)') = (b' ->1 a')
15 nomcon3 304 . 2 ((a v b) ==3 b) = (b' ==4 (a v b)')
16 i2i1 267 . 2 (a ->2 b) = (b' ->1 a')
1714, 15, 163tr1 63 1 ((a v b) ==3 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   ==3 wid3 20   ==4 wid4 21
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-id3 52  df-id4 53  df-le1 130  df-le2 131
This theorem is referenced by:  nom62  339
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