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Theorem salem1 837
Description: Lemma for attempt at Sasaki algebra.
Assertion
Ref Expression
salem1 (((a' ->1 b) v b) ->1 b) = (a ->2 b)
Proof of Theorem salem1
StepHypRef Expression
1 u1lemob 630 . . . . . 6 ((a' ->1 b) v b) = (a'' v b)
21ax-r4 37 . . . . 5 ((a' ->1 b) v b)' = (a'' v b)'
3 anor1 88 . . . . . 6 (a' ^ b') = (a'' v b)'
43ax-r1 35 . . . . 5 (a'' v b)' = (a' ^ b')
52, 4ax-r2 36 . . . 4 ((a' ->1 b) v b)' = (a' ^ b')
61ran 78 . . . . 5 (((a' ->1 b) v b) ^ b) = ((a'' v b) ^ b)
7 ax-a2 31 . . . . . . 7 (a'' v b) = (b v a'')
87ran 78 . . . . . 6 ((a'' v b) ^ b) = ((b v a'') ^ b)
9 ancom 74 . . . . . 6 ((b v a'') ^ b) = (b ^ (b v a''))
108, 9ax-r2 36 . . . . 5 ((a'' v b) ^ b) = (b ^ (b v a''))
11 anabs 121 . . . . 5 (b ^ (b v a'')) = b
126, 10, 113tr 65 . . . 4 (((a' ->1 b) v b) ^ b) = b
135, 122or 72 . . 3 (((a' ->1 b) v b)' v (((a' ->1 b) v b) ^ b)) = ((a' ^ b') v b)
14 ax-a2 31 . . 3 ((a' ^ b') v b) = (b v (a' ^ b'))
1513, 14ax-r2 36 . 2 (((a' ->1 b) v b)' v (((a' ->1 b) v b) ^ b)) = (b v (a' ^ b'))
16 df-i1 44 . 2 (((a' ->1 b) v b) ->1 b) = (((a' ->1 b) v b)' v (((a' ->1 b) v b) ^ b))
17 df-i2 45 . 2 (a ->2 b) = (b v (a' ^ b'))
1815, 16, 173tr1 63 1 (((a' ->1 b) v b) ->1 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
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-i1 44  df-i2 45  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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