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Theorem u2lem7n 775
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u2lem7n (a2 (a2 b)) = (((ab) ∩ (ab)) ∩ b )

Proof of Theorem u2lem7n
StepHypRef Expression
1 u2lem7 773 . . 3 (a2 (a2 b)) = (((ab ) ∪ (ab )) ∪ b)
2 ax-a2 31 . . . . . . 7 ((ab ) ∪ (ab )) = ((ab ) ∪ (ab ))
3 anor3 90 . . . . . . . 8 (ab ) = (ab)
4 anor1 88 . . . . . . . 8 (ab ) = (ab)
53, 42or 72 . . . . . . 7 ((ab ) ∪ (ab )) = ((ab) ∪ (ab) )
62, 5ax-r2 36 . . . . . 6 ((ab ) ∪ (ab )) = ((ab) ∪ (ab) )
7 oran3 93 . . . . . 6 ((ab) ∪ (ab) ) = ((ab) ∩ (ab))
86, 7ax-r2 36 . . . . 5 ((ab ) ∪ (ab )) = ((ab) ∩ (ab))
98ax-r5 38 . . . 4 (((ab ) ∪ (ab )) ∪ b) = (((ab) ∩ (ab))b)
10 oran2 92 . . . 4 (((ab) ∩ (ab))b) = (((ab) ∩ (ab)) ∩ b )
119, 10ax-r2 36 . . 3 (((ab ) ∪ (ab )) ∪ b) = (((ab) ∩ (ab)) ∩ b )
121, 11ax-r2 36 . 2 (a2 (a2 b)) = (((ab) ∩ (ab)) ∩ b )
1312con2 67 1 (a2 (a2 b)) = (((ab) ∩ (ab)) ∩ b )
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →2 wi2 13 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-i2 45  df-le1 130  df-le2 131 This theorem is referenced by:  u2lem8  782
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