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Theorem u5lemoa 624
Description: Lemma for relevance implication study.
Assertion
Ref Expression
u5lemoa ((a ->5 b) v a) = (a v ((a' ^ b) v (a' ^ b')))
Proof of Theorem u5lemoa
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 a) = ((((a ^ b) v (a' ^ b)) v (a' ^ b')) v a)
3 ax-a2 31 . . 3 ((((a ^ b) v (a' ^ b)) v (a' ^ b')) v a) = (a v (((a ^ b) v (a' ^ b)) v (a' ^ b')))
4 ax-a3 32 . . . . 5 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = ((a ^ b) v ((a' ^ b) v (a' ^ b')))
54lor 70 . . . 4 (a v (((a ^ b) v (a' ^ b)) v (a' ^ b'))) = (a v ((a ^ b) v ((a' ^ b) v (a' ^ b'))))
6 ax-a3 32 . . . . . 6 ((a v (a ^ b)) v ((a' ^ b) v (a' ^ b'))) = (a v ((a ^ b) v ((a' ^ b) v (a' ^ b'))))
76ax-r1 35 . . . . 5 (a v ((a ^ b) v ((a' ^ b) v (a' ^ b')))) = ((a v (a ^ b)) v ((a' ^ b) v (a' ^ b')))
8 orabs 120 . . . . . 6 (a v (a ^ b)) = a
98ax-r5 38 . . . . 5 ((a v (a ^ b)) v ((a' ^ b) v (a' ^ b'))) = (a v ((a' ^ b) v (a' ^ b')))
107, 9ax-r2 36 . . . 4 (a v ((a ^ b) v ((a' ^ b) v (a' ^ b')))) = (a v ((a' ^ b) v (a' ^ b')))
115, 10ax-r2 36 . . 3 (a v (((a ^ b) v (a' ^ b)) v (a' ^ b'))) = (a v ((a' ^ b) v (a' ^ b')))
123, 11ax-r2 36 . 2 ((((a ^ b) v (a' ^ b)) v (a' ^ b')) v a) = (a v ((a' ^ b) v (a' ^ b')))
132, 12ax-r2 36 1 ((a ->5 b) v a) = (a v ((a' ^ b) v (a' ^ 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-r5 38
This theorem depends on definitions:  df-a 40  df-i5 48
This theorem is referenced by:  u5lemnana  649
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