Theorem i5lei3 341

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

Assertion

Ref	Expression
i5lei3	(a ->5 b) =< (a ->3 b)

Proof of Theorem i5lei3

Step	Hyp	Ref	Expression
1		leor 151	. . . 4 b =< (a_|_ v b)
2	1	lelan 159	. . 3 (a ^ b) =< (a ^ (a_|_ v b))
3	2	leror 144	. 2 ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_))) =< ((a ^ (a_|_ v b)) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
4		df-i5 47	. . 3 (a ->5 b) = (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_))
5		ax-a3 31	. . 3 (((a ^ b) v (a_|_ ^ b)) v (a_|_ ^ b_|_)) = ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
6	4, 5	ax-r2 35	. 2 (a ->5 b) = ((a ^ b) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
7		df-i3 45	. . 3 (a ->3 b) = (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b)))
8		ax-a2 30	. . 3 (((a_|_ ^ b) v (a_|_ ^ b_|_)) v (a ^ (a_|_ v b))) = ((a ^ (a_|_ v b)) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
9	7, 8	ax-r2 35	. 2 (a ->3 b) = ((a ^ (a_|_ v b)) v ((a_|_ ^ b) v (a_|_ ^ b_|_)))
10	3, 6, 9	le3tr1 132	1 (a ->5 b) =< (a ->3 b)

Syntax hints:   =< wle 2  _|_wn 4   v wo 6   ^ wa 7   ->3 wi3 15   ->5 wi5 17

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-i3 45  df-i5 47  df-le1 122  df-le2 123

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