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Theorem u3lemoa 622
 Description: Lemma for Kalmbach implication study.
Assertion
Ref Expression
u3lemoa ((a3 b) ∪ a) = (a ∪ ((ab) ∪ (ab )))

Proof of Theorem u3lemoa
StepHypRef Expression
1 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
21ax-r5 38 . 2 ((a3 b) ∪ a) = ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ a)
3 ax-a3 32 . . 3 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ a) = (((ab) ∪ (ab )) ∪ ((a ∩ (ab)) ∪ a))
4 lea 160 . . . . . 6 (a ∩ (ab)) ≤ a
54df-le2 131 . . . . 5 ((a ∩ (ab)) ∪ a) = a
65lor 70 . . . 4 (((ab) ∪ (ab )) ∪ ((a ∩ (ab)) ∪ a)) = (((ab) ∪ (ab )) ∪ a)
7 ax-a2 31 . . . 4 (((ab) ∪ (ab )) ∪ a) = (a ∪ ((ab) ∪ (ab )))
86, 7ax-r2 36 . . 3 (((ab) ∪ (ab )) ∪ ((a ∩ (ab)) ∪ a)) = (a ∪ ((ab) ∪ (ab )))
93, 8ax-r2 36 . 2 ((((ab) ∪ (ab )) ∪ (a ∩ (ab))) ∪ a) = (a ∪ ((ab) ∪ (ab )))
102, 9ax-r2 36 1 ((a3 b) ∪ 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-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r5 38 This theorem depends on definitions:  df-a 40  df-i3 46  df-le1 130  df-le2 131 This theorem is referenced by:  u3lemnana  647
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