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Theorem nom23 316
Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom23 (a ==3 (a ^ b)) = (a ->1 b)
Proof of Theorem nom23
StepHypRef Expression
1 le1 146 . . . 4 (a' v (a ^ b)) =< 1
2 df-t 41 . . . . 5 1 = (a v a')
3 anabs 121 . . . . . . . 8 (a' ^ (a' v b')) = a'
43ax-r1 35 . . . . . . 7 a' = (a' ^ (a' v b'))
5 oran3 93 . . . . . . . 8 (a' v b') = (a ^ b)'
65lan 77 . . . . . . 7 (a' ^ (a' v b')) = (a' ^ (a ^ b)')
74, 6ax-r2 36 . . . . . 6 a' = (a' ^ (a ^ b)')
87lor 70 . . . . 5 (a v a') = (a v (a' ^ (a ^ b)'))
92, 8ax-r2 36 . . . 4 1 = (a v (a' ^ (a ^ b)'))
101, 9lbtr 139 . . 3 (a' v (a ^ b)) =< (a v (a' ^ (a ^ b)'))
1110df2le2 136 . 2 ((a' v (a ^ b)) ^ (a v (a' ^ (a ^ b)'))) = (a' v (a ^ b))
12 df-id3 52 . 2 (a ==3 (a ^ b)) = ((a' v (a ^ b)) ^ (a v (a' ^ (a ^ b)')))
13 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
1411, 12, 133tr1 63 1 (a ==3 (a ^ b)) = (a ->1 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->1 wi1 12   ==3 wid3 20
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-id3 52  df-le1 130  df-le2 131
This theorem is referenced by:  nom32  321  nom54  335
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