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Theorem u5lemoa 624
 Description: Lemma for relevance implication study.
Assertion
Ref Expression
u5lemoa ((a5 b) ∪ a) = (a ∪ ((ab) ∪ (ab )))

Proof of Theorem u5lemoa
StepHypRef Expression
1 df-i5 48 . . 3 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
21ax-r5 38 . 2 ((a5 b) ∪ a) = ((((ab) ∪ (ab)) ∪ (ab )) ∪ a)
3 ax-a2 31 . . 3 ((((ab) ∪ (ab)) ∪ (ab )) ∪ a) = (a ∪ (((ab) ∪ (ab)) ∪ (ab )))
4 ax-a3 32 . . . . 5 (((ab) ∪ (ab)) ∪ (ab )) = ((ab) ∪ ((ab) ∪ (ab )))
54lor 70 . . . 4 (a ∪ (((ab) ∪ (ab)) ∪ (ab ))) = (a ∪ ((ab) ∪ ((ab) ∪ (ab ))))
6 ax-a3 32 . . . . . 6 ((a ∪ (ab)) ∪ ((ab) ∪ (ab ))) = (a ∪ ((ab) ∪ ((ab) ∪ (ab ))))
76ax-r1 35 . . . . 5 (a ∪ ((ab) ∪ ((ab) ∪ (ab )))) = ((a ∪ (ab)) ∪ ((ab) ∪ (ab )))
8 orabs 120 . . . . . 6 (a ∪ (ab)) = a
98ax-r5 38 . . . . 5 ((a ∪ (ab)) ∪ ((ab) ∪ (ab ))) = (a ∪ ((ab) ∪ (ab )))
107, 9ax-r2 36 . . . 4 (a ∪ ((ab) ∪ ((ab) ∪ (ab )))) = (a ∪ ((ab) ∪ (ab )))
115, 10ax-r2 36 . . 3 (a ∪ (((ab) ∪ (ab)) ∪ (ab ))) = (a ∪ ((ab) ∪ (ab )))
123, 11ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ (ab )) ∪ a) = (a ∪ ((ab) ∪ (ab )))
132, 12ax-r2 36 1 ((a5 b) ∪ a) = (a ∪ ((ab) ∪ (ab )))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7   →5 wi5 16 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-i5 48 This theorem is referenced by:  u5lemnana  649
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