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Theorem ska8 236
Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA8.
Assertion
Ref Expression
ska8 ((a' ^ a) == ((a' ^ a) ^ b)) = 1

Proof of Theorem ska8
StepHypRef Expression
1 an0 108 . . . . 5 (b ^ 0) = 0
21ax-r1 35 . . . 4 0 = (b ^ 0)
3 ancom 74 . . . 4 (b ^ 0) = (0 ^ b)
42, 3ax-r2 36 . . 3 0 = (0 ^ b)
5 dff 101 . . . 4 0 = (a ^ a')
6 ancom 74 . . . 4 (a ^ a') = (a' ^ a)
75, 6ax-r2 36 . . 3 0 = (a' ^ a)
87ran 78 . . 3 (0 ^ b) = ((a' ^ a) ^ b)
94, 7, 83tr2 64 . 2 (a' ^ a) = ((a' ^ a) ^ b)
109bi1 118 1 ((a' ^ a) == ((a' ^ a) ^ b)) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   ^ wa 7  1wt 8  0wf 9
This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42
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