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Theorem i5lei3 349
Description: Relevance implication is l.e. Kalmbach implication.
Assertion
Ref Expression
i5lei3 (a ->5 b) =< (a ->3 b)
Proof of Theorem i5lei3
StepHypRef Expression
1 leor 159 . . . 4 b =< (a' v b)
21lelan 167 . . 3 (a ^ b) =< (a ^ (a' v b))
32leror 152 . 2 ((a ^ b) v ((a' ^ b) v (a' ^ b'))) =< ((a ^ (a' v b)) v ((a' ^ b) v (a' ^ b')))
4 df-i5 48 . . 3 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
5 ax-a3 32 . . 3 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = ((a ^ b) v ((a' ^ b) v (a' ^ b')))
64, 5ax-r2 36 . 2 (a ->5 b) = ((a ^ b) v ((a' ^ b) v (a' ^ b')))
7 df-i3 46 . . 3 (a ->3 b) = (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b)))
8 ax-a2 31 . . 3 (((a' ^ b) v (a' ^ b')) v (a ^ (a' v b))) = ((a ^ (a' v b)) v ((a' ^ b) v (a' ^ b')))
97, 8ax-r2 36 . 2 (a ->3 b) = ((a ^ (a' v b)) v ((a' ^ b) v (a' ^ b')))
103, 6, 9le3tr1 140 1 (a ->5 b) =< (a ->3 b)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->3 wi3 14   ->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-i3 46  df-i5 48  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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