Theorem i3ri3 538

Description: WQL (Weak Quantum Logic) rule. (Contributed by NM, 7-Nov-1997.)

Hypotheses

Ref	Expression
i3ri3.1	(a →3 b) = 1
i3ri3.2	(b →3 a) = 1

Assertion

Ref	Expression
i3ri3	((a →3 c) →3 (b →3 c)) = 1

Proof of Theorem i3ri3

Step	Hyp	Ref	Expression
1		i3ri3.1	. . . . . 6 (a →3 b) = 1
2	1	i3le 515	. . . . 5 a ≤ b
3		i3ri3.2	. . . . . 6 (b →3 a) = 1
4	3	i3le 515	. . . . 5 b ≤ a
5	2, 4	lebi 145	. . . 4 a = b
6	5	ri3 253	. . 3 (a →3 c) = (b →3 c)
7	6	bile 142	. 2 (a →3 c) ≤ (b →3 c)
8	7	lei3 246	1 ((a →3 c) →3 (b →3 c)) = 1

Syntax hints:   = wb 1  1wt 8   →₃ wi3 14

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)