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Theorem u2lem8 782
 Description: Lemma for unified implication study.
Assertion
Ref Expression
u2lem8 (a2 (a2 (a2 b))) = (a2 (a2 b))

Proof of Theorem u2lem8
StepHypRef Expression
1 df-i2 45 . 2 (a2 (a2 (a2 b))) = ((a2 (a2 b)) ∪ (a ∩ (a2 (a2 b)) ))
2 u2lem7 773 . . . 4 (a2 (a2 b)) = (((ab ) ∪ (ab )) ∪ b)
3 ax-a1 30 . . . . . . 7 a = a
43ax-r1 35 . . . . . 6 a = a
5 u2lem7n 775 . . . . . 6 (a2 (a2 b)) = (((ab) ∩ (ab)) ∩ b )
64, 52an 79 . . . . 5 (a ∩ (a2 (a2 b)) ) = (a ∩ (((ab) ∩ (ab)) ∩ b ))
7 an12 81 . . . . . 6 (a ∩ (((ab) ∩ (ab)) ∩ b )) = (((ab) ∩ (ab)) ∩ (ab ))
8 anass 76 . . . . . . 7 (((ab) ∩ (ab)) ∩ (ab )) = ((ab) ∩ ((ab) ∩ (ab )))
9 anor1 88 . . . . . . . . . . 11 (ab ) = (ab)
109lan 77 . . . . . . . . . 10 ((ab) ∩ (ab )) = ((ab) ∩ (ab) )
11 dff 101 . . . . . . . . . . 11 0 = ((ab) ∩ (ab) )
1211ax-r1 35 . . . . . . . . . 10 ((ab) ∩ (ab) ) = 0
1310, 12ax-r2 36 . . . . . . . . 9 ((ab) ∩ (ab )) = 0
1413lan 77 . . . . . . . 8 ((ab) ∩ ((ab) ∩ (ab ))) = ((ab) ∩ 0)
15 an0 108 . . . . . . . 8 ((ab) ∩ 0) = 0
1614, 15ax-r2 36 . . . . . . 7 ((ab) ∩ ((ab) ∩ (ab ))) = 0
178, 16ax-r2 36 . . . . . 6 (((ab) ∩ (ab)) ∩ (ab )) = 0
187, 17ax-r2 36 . . . . 5 (a ∩ (((ab) ∩ (ab)) ∩ b )) = 0
196, 18ax-r2 36 . . . 4 (a ∩ (a2 (a2 b)) ) = 0
202, 192or 72 . . 3 ((a2 (a2 b)) ∪ (a ∩ (a2 (a2 b)) )) = ((((ab ) ∪ (ab )) ∪ b) ∪ 0)
21 or0 102 . . . 4 ((((ab ) ∪ (ab )) ∪ b) ∪ 0) = (((ab ) ∪ (ab )) ∪ b)
222ax-r1 35 . . . 4 (((ab ) ∪ (ab )) ∪ b) = (a2 (a2 b))
2321, 22ax-r2 36 . . 3 ((((ab ) ∪ (ab )) ∪ b) ∪ 0) = (a2 (a2 b))
2420, 23ax-r2 36 . 2 ((a2 (a2 b)) ∪ (a ∩ (a2 (a2 b)) )) = (a2 (a2 b))
251, 24ax-r2 36 1 (a2 (a2 (a2 b))) = (a2 (a2 b))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →2 wi2 13 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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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