Theorem i3li3 521

Description: WQL (Weak Quantum Logic) rule.

Hypotheses

Ref	Expression
i3li3.1	(a ->3 b) = 1
i3li3.2	(b ->3 a) = 1

Assertion

Ref	Expression
i3li3	((c ->3 a) ->3 (c ->3 b)) = 1

Proof of Theorem i3li3

Step	Hyp	Ref	Expression
1		i3li3.1	. . . . . 6 (a ->3 b) = 1
2	1	i3le 497	. . . . 5 a =< b
3		i3li3.2	. . . . . 6 (b ->3 a) = 1
4	3	i3le 497	. . . . 5 b =< a
5	2, 4	lebi 137	. . . 4 a = b
6	5	li3 244	. . 3 (c ->3 a) = (c ->3 b)
7	6	bile 134	. 2 (c ->3 a) =< (c ->3 b)
8	7	lei3 238	1 ((c ->3 a) ->3 (c ->3 b)) = 1

Syntax hints:   = wb 1  1wt 9   ->3 wi3 15

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i3 45  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org