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Theorem oalem2 1006
 Description: Lemma.
Assertion
Ref Expression
oalem2 ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))) = (a2 b)

Proof of Theorem oalem2
StepHypRef Expression
1 ax-a2 31 . . . . . . 7 (bc) = (cb)
21ax-r4 37 . . . . . 6 (bc) = (cb)
3 ancom 74 . . . . . 6 ((a2 b) ∩ (a2 c)) = ((a2 c) ∩ (a2 b))
42, 32or 72 . . . . 5 ((bc) ∪ ((a2 b) ∩ (a2 c))) = ((cb) ∪ ((a2 c) ∩ (a2 b)))
54lan 77 . . . 4 ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 c) ∩ ((cb) ∪ ((a2 c) ∩ (a2 b))))
6 oath1 1004 . . . 4 ((a2 c) ∩ ((cb) ∪ ((a2 c) ∩ (a2 b)))) = ((a2 c) ∩ (a2 b))
75, 6ax-r2 36 . . 3 ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) = ((a2 c) ∩ (a2 b))
87lor 70 . 2 ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))) = ((a2 b) ∪ ((a2 c) ∩ (a2 b)))
9 ancom 74 . . 3 ((a2 c) ∩ (a2 b)) = ((a2 b) ∩ (a2 c))
109lor 70 . 2 ((a2 b) ∪ ((a2 c) ∩ (a2 b))) = ((a2 b) ∪ ((a2 b) ∩ (a2 c)))
11 orabs 120 . 2 ((a2 b) ∪ ((a2 b) ∩ (a2 c))) = (a2 b)
128, 10, 113tr 65 1 ((a2 b) ∪ ((a2 c) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c))))) = (a2 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  ax-3oa 998 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le1 130  df-le2 131 This theorem is referenced by: (None)
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