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Theorem ska15 244
 Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA15. (Contributed by NM, 2-Nov-1997.)
Assertion
Ref Expression
ska15 ((a3 b) ∪ (ab)) = 1

Proof of Theorem ska15
StepHypRef Expression
1 df-i3 46 . . 3 (a3 b) = (((ab) ∪ (ab )) ∪ (a ∩ (ab)))
2 ax-a2 31 . . . . . 6 ((ab) ∪ (ab )) = ((ab ) ∪ (ab))
3 lea 160 . . . . . . 7 (ab ) ≤ a
4 lear 161 . . . . . . 7 (ab) ≤ b
53, 4le2or 168 . . . . . 6 ((ab ) ∪ (ab)) ≤ (ab)
62, 5bltr 138 . . . . 5 ((ab) ∪ (ab )) ≤ (ab)
7 lear 161 . . . . 5 (a ∩ (ab)) ≤ (ab)
86, 7le2or 168 . . . 4 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) ≤ ((ab) ∪ (ab))
9 oridm 110 . . . 4 ((ab) ∪ (ab)) = (ab)
108, 9lbtr 139 . . 3 (((ab) ∪ (ab )) ∪ (a ∩ (ab))) ≤ (ab)
111, 10bltr 138 . 2 (a3 b) ≤ (ab)
1211sklem 230 1 ((a3 b) ∪ (ab)) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  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 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i3 46  df-le1 130  df-le2 131 This theorem is referenced by:  skmp3  245  mccune3  248
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