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Theorem omla 447
 Description: Orthomodular law. (Contributed by NM, 7-Nov-1997.)
Assertion
Ref Expression
omla (a ∩ (a ∪ (ab))) = (ab)

Proof of Theorem omla
StepHypRef Expression
1 df-a 40 . . . . . . 7 (a ∩ (ab )) = (a ∪ (ab ) )
2 df-a 40 . . . . . . . . . 10 (ab) = (ab )
32ax-r1 35 . . . . . . . . 9 (ab ) = (ab)
43lor 70 . . . . . . . 8 (a ∪ (ab ) ) = (a ∪ (ab))
54ax-r4 37 . . . . . . 7 (a ∪ (ab ) ) = (a ∪ (ab))
61, 5ax-r2 36 . . . . . 6 (a ∩ (ab )) = (a ∪ (ab))
76ax-r1 35 . . . . 5 (a ∪ (ab)) = (a ∩ (ab ))
87lor 70 . . . 4 (a ∪ (a ∪ (ab)) ) = (a ∪ (a ∩ (ab )))
9 omln 446 . . . 4 (a ∪ (a ∩ (ab ))) = (ab )
108, 9ax-r2 36 . . 3 (a ∪ (a ∪ (ab)) ) = (ab )
11 df-a 40 . . . 4 (a ∩ (a ∪ (ab))) = (a ∪ (a ∪ (ab)) )
1211con2 67 . . 3 (a ∩ (a ∪ (ab))) = (a ∪ (a ∪ (ab)) )
132con2 67 . . 3 (ab) = (ab )
1410, 12, 133tr1 63 . 2 (a ∩ (a ∪ (ab))) = (ab)
1514con1 66 1 (a ∩ (a ∪ (ab))) = (ab)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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  1082
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