Theorem i5lei2 340

Description: Relevance implication is l.e. Dishkant implication.

Assertion

Ref	Expression
i5lei2	(a ->5 b) =< (a ->2 b)

Proof of Theorem i5lei2

Step	Hyp	Ref	Expression
1		lear 153	. . . 4 (a ^ b) =< b
2		lear 153	. . . 4 (a_|_ ^ b) =< b
3	1, 2	lel2or 162	. . 3 ((a ^ b) v (a_|_ ^ b)) =< b
4	3	leror 144	. 2 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) =< (b v (a_|_ ^ b_|_))
5		df-i5 47	. 2 (a ->5 b) = (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))
6		df-i2 44	. 2 (a ->2 b) = (b v (a_|_ ^ b_|_))
7	4, 5, 6	le3tr1 132	1 (a ->5 b) =< (a ->2 b)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->2 wi2 14   ->5 wi5 17

This theorem is referenced by:  oago3.21x 872

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i2 44  df-i5 47  df-le1 122  df-le2 123

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