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Theorem u3lem3n 754
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u3lem3n (a3 (b3 a)) = (a ∩ ((ab) ∩ (ab )))

Proof of Theorem u3lem3n
StepHypRef Expression
1 u3lem3 751 . . 3 (a3 (b3 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 )) )
10 oran1 91 . . . 4 (a ∪ ((ab) ∩ (ab )) ) = (a ∩ ((ab) ∩ (ab )))
119, 10ax-r2 36 . . 3 (a ∪ ((ab) ∪ (ab ))) = (a ∩ ((ab) ∩ (ab )))
121, 11ax-r2 36 . 2 (a3 (b3 a)) = (a ∩ ((ab) ∩ (ab )))
1312con2 67 1 (a3 (b3 a)) = (a ∩ ((ab) ∩ (ab )))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →3 wi3 14 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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133 This theorem is referenced by: (None)
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