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Theorem nom53 334
 Description: Part of Lemma 3.3(15) from "Non-Orthomodular Models..." paper. (Contributed by NM, 7-Feb-1999.)
Assertion
Ref Expression
nom53 ((ab) ≡3 b) = (a2 b)

Proof of Theorem nom53
StepHypRef Expression
1 ancom 74 . . . . . . . 8 (ba ) = (ab )
2 anor3 90 . . . . . . . 8 (ab ) = (ab)
31, 2ax-r2 36 . . . . . . 7 (ba ) = (ab)
43ax-r1 35 . . . . . 6 (ab) = (ba )
54lor 70 . . . . 5 (b ∪ (ab) ) = (b ∪ (ba ))
64ax-r4 37 . . . . . 6 (ab) = (ba )
74lan 77 . . . . . 6 (b ∩ (ab) ) = (b ∩ (ba ))
86, 72or 72 . . . . 5 ((ab) ∪ (b ∩ (ab) )) = ((ba ) ∪ (b ∩ (ba )))
95, 82an 79 . . . 4 ((b ∪ (ab) ) ∩ ((ab) ∪ (b ∩ (ab) ))) = ((b ∪ (ba )) ∩ ((ba ) ∪ (b ∩ (ba ))))
10 df-id4 53 . . . 4 (b4 (ab) ) = ((b ∪ (ab) ) ∩ ((ab) ∪ (b ∩ (ab) )))
11 df-id4 53 . . . 4 (b4 (ba )) = ((b ∪ (ba )) ∩ ((ba ) ∪ (b ∩ (ba ))))
129, 10, 113tr1 63 . . 3 (b4 (ab) ) = (b4 (ba ))
13 nom24 317 . . 3 (b4 (ba )) = (b1 a )
1412, 13ax-r2 36 . 2 (b4 (ab) ) = (b1 a )
15 nomcon3 304 . 2 ((ab) ≡3 b) = (b4 (ab) )
16 i2i1 267 . 2 (a2 b) = (b1 a )
1714, 15, 163tr1 63 1 ((ab) ≡3 b) = (a2 b)
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →1 wi1 12   →2 wi2 13   ≡3 wid3 20   ≡4 wid4 21 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-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-id1 50  df-id2 51  df-id3 52  df-id4 53  df-le1 130  df-le2 131 This theorem is referenced by:  nom62  339
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