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Theorem omla 447
Description: Orthomodular law.
Assertion
Ref Expression
omla (a ^ (a' v (a ^ b))) = (a ^ b)
Proof of Theorem omla
StepHypRef Expression
1 df-a 40 . . . . . . 7 (a ^ (a' v b')) = (a' v (a' v b')')'
2 df-a 40 . . . . . . . . . 10 (a ^ b) = (a' v b')'
32ax-r1 35 . . . . . . . . 9 (a' v b')' = (a ^ b)
43lor 70 . . . . . . . 8 (a' v (a' v b')') = (a' v (a ^ b))
54ax-r4 37 . . . . . . 7 (a' v (a' v b')')' = (a' v (a ^ b))'
61, 5ax-r2 36 . . . . . 6 (a ^ (a' v b')) = (a' v (a ^ b))'
76ax-r1 35 . . . . 5 (a' v (a ^ b))' = (a ^ (a' v b'))
87lor 70 . . . 4 (a' v (a' v (a ^ b))') = (a' v (a ^ (a' v b')))
9 omln 446 . . . 4 (a' v (a ^ (a' v b'))) = (a' v b')
108, 9ax-r2 36 . . 3 (a' v (a' v (a ^ b))') = (a' v b')
11 df-a 40 . . . 4 (a ^ (a' v (a ^ b))) = (a' v (a' v (a ^ b))')'
1211con2 67 . . 3 (a ^ (a' v (a ^ b)))' = (a' v (a' v (a ^ b))')
132con2 67 . . 3 (a ^ b)' = (a' v b')
1410, 12, 133tr1 63 . 2 (a ^ (a' v (a ^ b)))' = (a ^ b)'
1514con1 66 1 (a ^ (a' v (a ^ b))) = (a ^ b)
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7
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  ax-r3 439
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42
This theorem is referenced by:  omlan  448  oml5a  450  gsth2  490  oa3-2to2s  990  lem4.6.2e1  1080
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