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Theorem u5lemona 629
 Description: Lemma for relevance implication study. (Contributed by NM, 15-Dec-1997.)
Assertion
Ref Expression
u5lemona ((a5 b) ∪ a ) = (a ∪ (ab))

Proof of Theorem u5lemona
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-a3 32 . . . 4 (((ab) ∪ (ab)) ∪ (ab )) = ((ab) ∪ ((ab) ∪ (ab )))
43ax-r5 38 . . 3 ((((ab) ∪ (ab)) ∪ (ab )) ∪ a ) = (((ab) ∪ ((ab) ∪ (ab ))) ∪ a )
5 ax-a3 32 . . . 4 (((ab) ∪ ((ab) ∪ (ab ))) ∪ a ) = ((ab) ∪ (((ab) ∪ (ab )) ∪ a ))
6 lea 160 . . . . . . . 8 (ab) ≤ a
7 lea 160 . . . . . . . 8 (ab ) ≤ a
86, 7lel2or 170 . . . . . . 7 ((ab) ∪ (ab )) ≤ a
98df-le2 131 . . . . . 6 (((ab) ∪ (ab )) ∪ a ) = a
109lor 70 . . . . 5 ((ab) ∪ (((ab) ∪ (ab )) ∪ a )) = ((ab) ∪ a )
11 ax-a2 31 . . . . 5 ((ab) ∪ a ) = (a ∪ (ab))
1210, 11ax-r2 36 . . . 4 ((ab) ∪ (((ab) ∪ (ab )) ∪ a )) = (a ∪ (ab))
135, 12ax-r2 36 . . 3 (((ab) ∪ ((ab) ∪ (ab ))) ∪ a ) = (a ∪ (ab))
144, 13ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ (ab )) ∪ a ) = (a ∪ (ab))
152, 14ax-r2 36 1 ((a5 b) ∪ a ) = (a ∪ (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-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-t 41  df-f 42  df-i5 48  df-le1 130  df-le2 131 This theorem is referenced by:  u5lemnaa  644  u5lem5  765
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