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Theorem u5lemnonb 679
 Description: Lemma for relevance implication study. (Contributed by NM, 16-Dec-1997.)
Assertion
Ref Expression
u5lemnonb ((a5 b)b ) = ((ab ) ∩ (ab ))

Proof of Theorem u5lemnonb
StepHypRef Expression
1 u5lemab 614 . . . 4 ((a5 b) ∩ b) = ((ab) ∪ (ab))
2 ax-a2 31 . . . . 5 ((ab) ∪ (ab)) = ((ab) ∪ (ab))
3 anor2 89 . . . . . 6 (ab) = (ab )
4 df-a 40 . . . . . 6 (ab) = (ab )
53, 42or 72 . . . . 5 ((ab) ∪ (ab)) = ((ab ) ∪ (ab ) )
62, 5ax-r2 36 . . . 4 ((ab) ∪ (ab)) = ((ab ) ∪ (ab ) )
71, 6ax-r2 36 . . 3 ((a5 b) ∩ b) = ((ab ) ∪ (ab ) )
8 df-a 40 . . 3 ((a5 b) ∩ b) = ((a5 b)b )
9 oran3 93 . . 3 ((ab ) ∪ (ab ) ) = ((ab ) ∩ (ab ))
107, 8, 93tr2 64 . 2 ((a5 b)b ) = ((ab ) ∩ (ab ))
1110con1 66 1 ((a5 b)b ) = ((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-a4 33  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  df-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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