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Theorem u5lemob 634
 Description: Lemma for relevance implication study. (Contributed by NM, 15-Dec-1997.)
Assertion
Ref Expression
u5lemob ((a5 b) ∪ b) = ((ab ) ∪ b)

Proof of Theorem u5lemob
StepHypRef Expression
1 df-i5 48 . . 3 (a5 b) = (((ab) ∪ (ab)) ∪ (ab ))
21ax-r5 38 . 2 ((a5 b) ∪ b) = ((((ab) ∪ (ab)) ∪ (ab )) ∪ b)
3 ax-a3 32 . . 3 ((((ab) ∪ (ab)) ∪ (ab )) ∪ b) = (((ab) ∪ (ab)) ∪ ((ab ) ∪ b))
4 lear 161 . . . . . 6 (ab) ≤ b
5 lear 161 . . . . . 6 (ab) ≤ b
64, 5lel2or 170 . . . . 5 ((ab) ∪ (ab)) ≤ b
7 leor 159 . . . . 5 b ≤ ((ab ) ∪ b)
86, 7letr 137 . . . 4 ((ab) ∪ (ab)) ≤ ((ab ) ∪ b)
98df-le2 131 . . 3 (((ab) ∪ (ab)) ∪ ((ab ) ∪ b)) = ((ab ) ∪ b)
103, 9ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ (ab )) ∪ b) = ((ab ) ∪ b)
112, 10ax-r2 36 1 ((a5 b) ∪ b) = ((ab ) ∪ b)
 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:  u5lemnanb  659
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