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Theorem nom25 318
 Description: Part of Lemma 3.3(14) from "Non-Orthomodular Models..." paper.
Assertion
Ref Expression
nom25 (a ≡ (ab)) = (a1 b)

Proof of Theorem nom25
StepHypRef Expression
1 anass 76 . . . . . 6 ((aa) ∩ b) = (a ∩ (ab))
21ax-r1 35 . . . . 5 (a ∩ (ab)) = ((aa) ∩ b)
3 anidm 111 . . . . . 6 (aa) = a
43ran 78 . . . . 5 ((aa) ∩ b) = (ab)
52, 4ax-r2 36 . . . 4 (a ∩ (ab)) = (ab)
6 oran3 93 . . . . . . 7 (ab ) = (ab)
76lan 77 . . . . . 6 (a ∩ (ab )) = (a ∩ (ab) )
87ax-r1 35 . . . . 5 (a ∩ (ab) ) = (a ∩ (ab ))
9 anabs 121 . . . . 5 (a ∩ (ab )) = a
108, 9ax-r2 36 . . . 4 (a ∩ (ab) ) = a
115, 102or 72 . . 3 ((a ∩ (ab)) ∪ (a ∩ (ab) )) = ((ab) ∪ a )
12 ax-a2 31 . . 3 ((ab) ∪ a ) = (a ∪ (ab))
1311, 12ax-r2 36 . 2 ((a ∩ (ab)) ∪ (a ∩ (ab) )) = (a ∪ (ab))
14 dfb 94 . 2 (a ≡ (ab)) = ((a ∩ (ab)) ∪ (a ∩ (ab) ))
15 df-i1 44 . 2 (a1 b) = (a ∪ (ab))
1613, 14, 153tr1 63 1 (a ≡ (ab)) = (a1 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7   →1 wi1 12 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-b 39  df-a 40  df-t 41  df-f 42  df-i1 44 This theorem is referenced by:  nom35  324  nom55  336
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