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Theorem i32i3 540
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
StepHypRef Expression
1 i32i3.1 . . . . . 6 (a ->3 b) = 1
21i3le 515 . . . . 5 a =< b
3 i32i3.2 . . . . . 6 (b ->3 a) = 1
43i3le 515 . . . . 5 b =< a
52, 4lebi 145 . . . 4 a = b
6 i32i3.3 . . . . . 6 (c ->3 d) = 1
76i3le 515 . . . . 5 c =< d
8 i32i3.4 . . . . . 6 (d ->3 c) = 1
98i3le 515 . . . . 5 d =< c
107, 9lebi 145 . . . 4 c = d
115, 102i3 254 . . 3 (a ->3 c) = (b ->3 d)
1211bile 142 . 2 (a ->3 c) =< (b ->3 d)
1312lei3 246 1 ((a ->3 c) ->3 (b ->3 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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