Theorem i32i3 522

Description: WQL (Weak Quantum Logic) rule.

Hypotheses

Ref	Expression
i32i3.1	(a →3 b) = 1
i32i3.2	(b →3 a) = 1
i32i3.3	(c →3 d) = 1
i32i3.4	(d →3 c) = 1

Assertion

Ref	Expression
i32i3	((a →3 c) →3 (b →3 d)) = 1

Proof of Theorem i32i3

Step	Hyp	Ref	Expression
1		i32i3.1	. . . . . 6 (a →3 b) = 1
2	1	i3le 497	. . . . 5 a ≤ b
3		i32i3.2	. . . . . 6 (b →3 a) = 1
4	3	i3le 497	. . . . 5 b ≤ a
5	2, 4	lebi 137	. . . 4 a = b
6		i32i3.3	. . . . . 6 (c →3 d) = 1
7	6	i3le 497	. . . . 5 c ≤ d
8		i32i3.4	. . . . . 6 (d →3 c) = 1
9	8	i3le 497	. . . . 5 d ≤ c
10	7, 9	lebi 137	. . . 4 c = d
11	5, 10	2i3 246	. . 3 (a →3 c) = (b →3 d)
12	11	bile 134	. 2 (a →3 c) ≤ (b →3 d)
13	12	lei3 238	1 ((a →3 c) →3 (b →3 d)) = 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

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