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Theorem u5lemnana 649
 Description: Lemma for relevance implication study.
Assertion
Ref Expression
u5lemnana ((a5 b)a ) = (a ∩ ((ab) ∩ (ab )))

Proof of Theorem u5lemnana
StepHypRef Expression
1 u5lemoa 624 . . . 4 ((a5 b) ∪ a) = (a ∪ ((ab) ∪ (ab )))
2 ax-a2 31 . . . . . 6 ((ab) ∪ (ab )) = ((ab ) ∪ (ab))
3 anor3 90 . . . . . . . 8 (ab ) = (ab)
4 anor2 89 . . . . . . . 8 (ab) = (ab )
53, 42or 72 . . . . . . 7 ((ab ) ∪ (ab)) = ((ab) ∪ (ab ) )
6 oran3 93 . . . . . . 7 ((ab) ∪ (ab ) ) = ((ab) ∩ (ab ))
75, 6ax-r2 36 . . . . . 6 ((ab ) ∪ (ab)) = ((ab) ∩ (ab ))
82, 7ax-r2 36 . . . . 5 ((ab) ∪ (ab )) = ((ab) ∩ (ab ))
98lor 70 . . . 4 (a ∪ ((ab) ∪ (ab ))) = (a ∪ ((ab) ∩ (ab )) )
101, 9ax-r2 36 . . 3 ((a5 b) ∪ a) = (a ∪ ((ab) ∩ (ab )) )
11 oran 87 . . 3 ((a5 b) ∪ a) = ((a5 b)a )
12 oran1 91 . . 3 (a ∪ ((ab) ∩ (ab )) ) = (a ∩ ((ab) ∩ (ab )))
1310, 11, 123tr2 64 . 2 ((a5 b)a ) = (a ∩ ((ab) ∩ (ab )))
1413con1 66 1 ((a5 b)a ) = (a ∩ ((ab) ∩ (ab )))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →5 wi5 16 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-i5 48 This theorem is referenced by: (None)
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