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Theorem i32i3 540
 Description: WQL (Weak Quantum Logic) rule.
Hypotheses
Ref Expression
i32i3.1 (a3 b) = 1
i32i3.2 (b3 a) = 1
i32i3.3 (c3 d) = 1
i32i3.4 (d3 c) = 1
Assertion
Ref Expression
i32i3 ((a3 c) →3 (b3 d)) = 1

Proof of Theorem i32i3
StepHypRef Expression
1 i32i3.1 . . . . . 6 (a3 b) = 1
21i3le 515 . . . . 5 ab
3 i32i3.2 . . . . . 6 (b3 a) = 1
43i3le 515 . . . . 5 ba
52, 4lebi 145 . . . 4 a = b
6 i32i3.3 . . . . . 6 (c3 d) = 1
76i3le 515 . . . . 5 cd
8 i32i3.4 . . . . . 6 (d3 c) = 1
98i3le 515 . . . . 5 dc
107, 9lebi 145 . . . 4 c = d
115, 102i3 254 . . 3 (a3 c) = (b3 d)
1211bile 142 . 2 (a3 c) ≤ (b3 d)
1312lei3 246 1 ((a3 c) →3 (b3 d)) = 1
 Colors of variables: term 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)
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