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Theorem nom25 318
Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom25 (a == (a ^ b)) = (a ->1 b)
Proof of Theorem nom25
StepHypRef Expression
1 anass 76 . . . . . 6 ((a ^ a) ^ b) = (a ^ (a ^ b))
21ax-r1 35 . . . . 5 (a ^ (a ^ b)) = ((a ^ a) ^ b)
3 anidm 111 . . . . . 6 (a ^ a) = a
43ran 78 . . . . 5 ((a ^ a) ^ b) = (a ^ b)
52, 4ax-r2 36 . . . 4 (a ^ (a ^ b)) = (a ^ b)
6 oran3 93 . . . . . . 7 (a' v b') = (a ^ b)'
76lan 77 . . . . . 6 (a' ^ (a' v b')) = (a' ^ (a ^ b)')
87ax-r1 35 . . . . 5 (a' ^ (a ^ b)') = (a' ^ (a' v b'))
9 anabs 121 . . . . 5 (a' ^ (a' v b')) = a'
108, 9ax-r2 36 . . . 4 (a' ^ (a ^ b)') = a'
115, 102or 72 . . 3 ((a ^ (a ^ b)) v (a' ^ (a ^ b)')) = ((a ^ b) v a')
12 ax-a2 31 . . 3 ((a ^ b) v a') = (a' v (a ^ b))
1311, 12ax-r2 36 . 2 ((a ^ (a ^ b)) v (a' ^ (a ^ b)')) = (a' v (a ^ b))
14 dfb 94 . 2 (a == (a ^ b)) = ((a ^ (a ^ b)) v (a' ^ (a ^ b)'))
15 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
1613, 14, 153tr1 63 1 (a == (a ^ b)) = (a ->1 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7   ->1 wi1 12
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-b 39  df-a 40  df-t 41  df-f 42  df-i1 44
This theorem is referenced by:  nom35  324  nom55  336
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