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Theorem ska13 241
Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA13.
Assertion
Ref Expression
ska13 ((a == b)' v (a' v b)) = 1
Proof of Theorem ska13
StepHypRef Expression
1 ledio 176 . . . . 5 ((a ^ b) v (a' ^ b')) =< (((a ^ b) v a') ^ ((a ^ b) v b'))
2 lea 160 . . . . 5 (((a ^ b) v a') ^ ((a ^ b) v b')) =< ((a ^ b) v a')
31, 2letr 137 . . . 4 ((a ^ b) v (a' ^ b')) =< ((a ^ b) v a')
4 ancom 74 . . . . . 6 (a ^ b) = (b ^ a)
5 lea 160 . . . . . 6 (b ^ a) =< b
64, 5bltr 138 . . . . 5 (a ^ b) =< b
76leror 152 . . . 4 ((a ^ b) v a') =< (b v a')
83, 7letr 137 . . 3 ((a ^ b) v (a' ^ b')) =< (b v a')
9 dfb 94 . . 3 (a == b) = ((a ^ b) v (a' ^ b'))
10 ax-a2 31 . . 3 (a' v b) = (b v a')
118, 9, 10le3tr1 140 . 2 (a == b) =< (a' v b)
1211sklem 230 1 ((a == b)' v (a' v b)) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   == tb 5   v wo 6   ^ wa 7  1wt 8
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
This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131
This theorem is referenced by: (None)
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