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Theorem ska13 241
 Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA13.
Assertion
Ref Expression
ska13 ((ab) ∪ (ab)) = 1

Proof of Theorem ska13
StepHypRef Expression
1 ledio 176 . . . . 5 ((ab) ∪ (ab )) ≤ (((ab) ∪ a ) ∩ ((ab) ∪ b ))
2 lea 160 . . . . 5 (((ab) ∪ a ) ∩ ((ab) ∪ b )) ≤ ((ab) ∪ a )
31, 2letr 137 . . . 4 ((ab) ∪ (ab )) ≤ ((ab) ∪ a )
4 ancom 74 . . . . . 6 (ab) = (ba)
5 lea 160 . . . . . 6 (ba) ≤ b
64, 5bltr 138 . . . . 5 (ab) ≤ b
76leror 152 . . . 4 ((ab) ∪ a ) ≤ (ba )
83, 7letr 137 . . 3 ((ab) ∪ (ab )) ≤ (ba )
9 dfb 94 . . 3 (ab) = ((ab) ∪ (ab ))
10 ax-a2 31 . . 3 (ab) = (ba )
118, 9, 10le3tr1 140 . 2 (ab) ≤ (ab)
1211sklem 230 1 ((ab) ∪ (ab)) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ≡ tb 5   ∪ 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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