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Theorem nom23 316
 Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper. (Contributed by NM, 7-Feb-1999.)
Assertion
Ref Expression
nom23 (a3 (ab)) = (a1 b)

Proof of Theorem nom23
StepHypRef Expression
1 le1 146 . . . 4 (a ∪ (ab)) ≤ 1
2 df-t 41 . . . . 5 1 = (aa )
3 anabs 121 . . . . . . . 8 (a ∩ (ab )) = a
43ax-r1 35 . . . . . . 7 a = (a ∩ (ab ))
5 oran3 93 . . . . . . . 8 (ab ) = (ab)
65lan 77 . . . . . . 7 (a ∩ (ab )) = (a ∩ (ab) )
74, 6ax-r2 36 . . . . . 6 a = (a ∩ (ab) )
87lor 70 . . . . 5 (aa ) = (a ∪ (a ∩ (ab) ))
92, 8ax-r2 36 . . . 4 1 = (a ∪ (a ∩ (ab) ))
101, 9lbtr 139 . . 3 (a ∪ (ab)) ≤ (a ∪ (a ∩ (ab) ))
1110df2le2 136 . 2 ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) ))) = (a ∪ (ab))
12 df-id3 52 . 2 (a3 (ab)) = ((a ∪ (ab)) ∩ (a ∪ (a ∩ (ab) )))
13 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
1411, 12, 133tr1 63 1 (a3 (ab)) = (a1 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →1 wi1 12   ≡3 wid3 20 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  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-i1 44  df-id3 52  df-le1 130  df-le2 131 This theorem is referenced by:  nom32  321  nom54  335
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