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Theorem nom13 310
Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom13 (a ->3 (a ^ b)) = (a ->1 b)
Proof of Theorem nom13
StepHypRef Expression
1 oran 87 . . . . . . . . 9 (a v (a ^ b)) = (a' ^ (a ^ b)')'
21ax-r1 35 . . . . . . . 8 (a' ^ (a ^ b)')' = (a v (a ^ b))
3 orabs 120 . . . . . . . 8 (a v (a ^ b)) = a
42, 3ax-r2 36 . . . . . . 7 (a' ^ (a ^ b)')' = a
54con3 68 . . . . . 6 (a' ^ (a ^ b)') = a'
65lor 70 . . . . 5 ((a' ^ (a ^ b)) v (a' ^ (a ^ b)')) = ((a' ^ (a ^ b)) v a')
7 lea 160 . . . . . 6 (a' ^ (a ^ b)) =< a'
87df-le2 131 . . . . 5 ((a' ^ (a ^ b)) v a') = a'
96, 8ax-r2 36 . . . 4 ((a' ^ (a ^ b)) v (a' ^ (a ^ b)')) = a'
109ax-r5 38 . . 3 (((a' ^ (a ^ b)) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = (a' v (a ^ (a' v (a ^ b))))
11 womaa 222 . . 3 (a' v (a ^ (a' v (a ^ b)))) = (a' v (a ^ b))
1210, 11ax-r2 36 . 2 (((a' ^ (a ^ b)) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b)))) = (a' v (a ^ b))
13 df-i3 46 . 2 (a ->3 (a ^ b)) = (((a' ^ (a ^ b)) v (a' ^ (a ^ b)')) v (a ^ (a' v (a ^ b))))
14 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
1512, 13, 143tr1 63 1 (a ->3 (a ^ b)) = (a ->1 b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->3 wi3 14
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-i3 46  df-le1 130  df-le2 131
This theorem is referenced by:  nom44  329  lem3.3.7i3e3  1068
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