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Theorem u5lemob 634
Description: Lemma for relevance implication study.
Assertion
Ref Expression
u5lemob ((a ->5 b) v b) = ((a' ^ b') v b)
Proof of Theorem u5lemob
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 . . . . . 6 (a ^ b) =< b
5 lear 161 . . . . . 6 (a' ^ b) =< b
64, 5lel2or 170 . . . . 5 ((a ^ b) v (a' ^ b)) =< b
7 leor 159 . . . . 5 b =< ((a' ^ b') v b)
86, 7letr 137 . . . 4 ((a ^ b) v (a' ^ b)) =< ((a' ^ b') v b)
98df-le2 131 . . 3 (((a ^ b) v (a' ^ b)) v ((a' ^ b') v b)) = ((a' ^ b') v b)
103, 9ax-r2 36 . 2 ((((a ^ b) v (a' ^ b)) v (a' ^ b')) v b) = ((a' ^ b') v b)
112, 10ax-r2 36 1 ((a ->5 b) v b) = ((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-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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