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Theorem oml6 488
Description: Orthomodular law.
Assertion
Ref Expression
oml6 (a v (b ^ (a' v b'))) = (a v b)
Proof of Theorem oml6
StepHypRef Expression
1 comor1 461 . . . 4 (a' v b') C a'
21comcom7 460 . . 3 (a' v b') C a
3 comor2 462 . . . 4 (a' v b') C b'
43comcom7 460 . . 3 (a' v b') C b
52, 4fh4c 478 . 2 (a v (b ^ (a' v b'))) = ((a v b) ^ (a v (a' v b')))
6 df-t 41 . . . . . 6 1 = (a v a')
76ax-r5 38 . . . . 5 (1 v b') = ((a v a') v b')
8 ax-a2 31 . . . . . 6 (1 v b') = (b' v 1)
9 or1 104 . . . . . 6 (b' v 1) = 1
108, 9ax-r2 36 . . . . 5 (1 v b') = 1
11 ax-a3 32 . . . . 5 ((a v a') v b') = (a v (a' v b'))
127, 10, 113tr2 64 . . . 4 1 = (a v (a' v b'))
1312ax-r1 35 . . 3 (a v (a' v b')) = 1
1413lan 77 . 2 ((a v b) ^ (a v (a' v b'))) = ((a v b) ^ 1)
15 an1 106 . 2 ((a v b) ^ 1) = (a v b)
165, 14, 153tr 65 1 (a v (b ^ (a' v b'))) = (a v b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  sa5  836
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