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Theorem nom22 315
Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom22 (a ==2 (a ^ b)) = (a ->1 b)
Proof of Theorem nom22
StepHypRef Expression
1 oran3 93 . . . . . . 7 (a' v b') = (a ^ b)'
21lor 70 . . . . . 6 (a v (a' v b')) = (a v (a ^ b)')
32ax-r1 35 . . . . 5 (a v (a ^ b)') = (a v (a' v b'))
4 or12 80 . . . . 5 (a v (a' v b')) = (a' v (a v b'))
53, 4ax-r2 36 . . . 4 (a v (a ^ b)') = (a' v (a v b'))
6 ax-a2 31 . . . . 5 ((a ^ b) v (a' ^ (a ^ b)')) = ((a' ^ (a ^ b)') v (a ^ b))
71lan 77 . . . . . . . 8 (a' ^ (a' v b')) = (a' ^ (a ^ b)')
87ax-r1 35 . . . . . . 7 (a' ^ (a ^ b)') = (a' ^ (a' v b'))
9 anabs 121 . . . . . . 7 (a' ^ (a' v b')) = a'
108, 9ax-r2 36 . . . . . 6 (a' ^ (a ^ b)') = a'
1110ax-r5 38 . . . . 5 ((a' ^ (a ^ b)') v (a ^ b)) = (a' v (a ^ b))
126, 11ax-r2 36 . . . 4 ((a ^ b) v (a' ^ (a ^ b)')) = (a' v (a ^ b))
135, 122an 79 . . 3 ((a v (a ^ b)') ^ ((a ^ b) v (a' ^ (a ^ b)'))) = ((a' v (a v b')) ^ (a' v (a ^ b)))
14 ancom 74 . . 3 ((a' v (a v b')) ^ (a' v (a ^ b))) = ((a' v (a ^ b)) ^ (a' v (a v b')))
15 lea 160 . . . . . 6 (a ^ b) =< a
16 leo 158 . . . . . 6 a =< (a v b')
1715, 16letr 137 . . . . 5 (a ^ b) =< (a v b')
1817lelor 166 . . . 4 (a' v (a ^ b)) =< (a' v (a v b'))
1918df2le2 136 . . 3 ((a' v (a ^ b)) ^ (a' v (a v b'))) = (a' v (a ^ b))
2013, 14, 193tr 65 . 2 ((a v (a ^ b)') ^ ((a ^ b) v (a' ^ (a ^ b)'))) = (a' v (a ^ b))
21 df-id2 51 . 2 (a ==2 (a ^ b)) = ((a v (a ^ b)') ^ ((a ^ b) v (a' ^ (a ^ b)')))
22 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
2320, 21, 223tr1 63 1 (a ==2 (a ^ b)) = (a ->1 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ==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-id2 51  df-le1 130  df-le2 131
This theorem is referenced by:  nom33  322  nom51  332
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