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Theorem i5lei1 347
Description: Relevance implication is l.e. Sasaki implication.
Assertion
Ref Expression
i5lei1 (a ->5 b) =< (a ->1 b)
Proof of Theorem i5lei1
StepHypRef Expression
1 ax-a3 32 . . . 4 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = ((a ^ b) v ((a' ^ b) v (a' ^ b')))
2 ax-a2 31 . . . 4 ((a ^ b) v ((a' ^ b) v (a' ^ b'))) = (((a' ^ b) v (a' ^ b')) v (a ^ b))
31, 2ax-r2 36 . . 3 (((a ^ b) v (a' ^ b)) v (a' ^ b')) = (((a' ^ b) v (a' ^ b')) v (a ^ b))
4 lea 160 . . . . 5 (a' ^ b) =< a'
5 lea 160 . . . . 5 (a' ^ b') =< a'
64, 5lel2or 170 . . . 4 ((a' ^ b) v (a' ^ b')) =< a'
76leror 152 . . 3 (((a' ^ b) v (a' ^ b')) v (a ^ b)) =< (a' v (a ^ b))
83, 7bltr 138 . 2 (((a ^ b) v (a' ^ b)) v (a' ^ b')) =< (a' v (a ^ b))
9 df-i5 48 . 2 (a ->5 b) = (((a ^ b) v (a' ^ b)) v (a' ^ b'))
10 df-i1 44 . 2 (a ->1 b) = (a' v (a ^ b))
118, 9, 10le3tr1 140 1 (a ->5 b) =< (a ->1 b)
Colors of variables: term
Syntax hints:   =< wle 2  'wn 4   v wo 6   ^ wa 7   ->1 wi1 12   ->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-i1 44  df-i5 48  df-le1 130  df-le2 131
This theorem is referenced by:  oago3.21x  890
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