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Theorem u5lemonb 639
Description: Lemma for relevance implication study.
Assertion
Ref Expression
u5lemonb ((a ->5 b) v b') = (((a ^ b) v (a' ^ b)) v b')
Proof of Theorem u5lemonb
StepHypRef Expression
1 df-i5 48 . . 3 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
21ax-r5 38 . 2 ((a ->5 b) v b') = ((((a ^ b) v (a' ^ b)) v (a' ^ b')) v b')
3 ax-a3 32 . . 3 ((((a ^ b) v (a' ^ b)) v (a' ^ b')) v b') = (((a ^ b) v (a' ^ b)) v ((a' ^ b') v b'))
4 lear 161 . . . . 5 (a' ^ b') =< b'
54df-le2 131 . . . 4 ((a' ^ b') v b') = b'
65lor 70 . . 3 (((a ^ b) v (a' ^ b)) v ((a' ^ b') v b')) = (((a ^ b) v (a' ^ b)) v b')
73, 6ax-r2 36 . 2 ((((a ^ b) v (a' ^ b)) v (a' ^ b')) v b') = (((a ^ b) v (a' ^ b)) v b')
82, 7ax-r2 36 1 ((a ->5 b) v b') = (((a ^ b) v (a' ^ b)) v b')
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v 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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