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Theorem u5lemonb 639
 Description: Lemma for relevance implication study.
Assertion
Ref Expression
u5lemonb ((a5 b) ∪ b ) = (((ab) ∪ (ab)) ∪ b )

Proof of Theorem u5lemonb
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 . . . . 5 (ab ) ≤ b
54df-le2 131 . . . 4 ((ab ) ∪ b ) = b
65lor 70 . . 3 (((ab) ∪ (ab)) ∪ ((ab ) ∪ b )) = (((ab) ∪ (ab)) ∪ b )
73, 6ax-r2 36 . 2 ((((ab) ∪ (ab)) ∪ (ab )) ∪ b ) = (((ab) ∪ (ab)) ∪ b )
82, 7ax-r2 36 1 ((a5 b) ∪ b ) = (((ab) ∪ (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-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-i5 48  df-le1 130  df-le2 131 This theorem is referenced by:  u5lemnab  654  u5lem3  753
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