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Theorem nom54 335
Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom54 ((a v b) ==4 b) = (a ->2 b)
Proof of Theorem nom54
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)'
43lor 70 . . . . . 6 (b'' v (b' ^ a')) = (b'' v (a v b)')
53ax-r4 37 . . . . . . . 8 (b' ^ a')' = (a v b)''
65lan 77 . . . . . . 7 (b'' ^ (b' ^ a')') = (b'' ^ (a v b)'')
76lor 70 . . . . . 6 (b' v (b'' ^ (b' ^ a')')) = (b' v (b'' ^ (a v b)''))
84, 72an 79 . . . . 5 ((b'' v (b' ^ a')) ^ (b' v (b'' ^ (b' ^ a')'))) = ((b'' v (a v b)') ^ (b' v (b'' ^ (a v b)'')))
9 df-id3 52 . . . . 5 (b' ==3 (b' ^ a')) = ((b'' v (b' ^ a')) ^ (b' v (b'' ^ (b' ^ a')')))
10 df-id3 52 . . . . 5 (b' ==3 (a v b)') = ((b'' v (a v b)') ^ (b' v (b'' ^ (a v b)'')))
118, 9, 103tr1 63 . . . 4 (b' ==3 (b' ^ a')) = (b' ==3 (a v b)')
1211ax-r1 35 . . 3 (b' ==3 (a v b)') = (b' ==3 (b' ^ a'))
13 nom23 316 . . 3 (b' ==3 (b' ^ a')) = (b' ->1 a')
1412, 13ax-r2 36 . 2 (b' ==3 (a v b)') = (b' ->1 a')
15 nomcon4 305 . 2 ((a v b) ==4 b) = (b' ==3 (a v b)')
16 i2i1 267 . 2 (a ->2 b) = (b' ->1 a')
1714, 15, 163tr1 63 1 ((a v b) ==4 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-a4 33  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-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:  nom61  338
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