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Theorem skr0 242
 Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KR0.
Hypotheses
Ref Expression
skr0.1 a = 1
skr0.2 (ab) = 1
Assertion
Ref Expression
skr0 b = 1

Proof of Theorem skr0
StepHypRef Expression
1 ax-a2 31 . . 3 (b ∪ 0) = (0 ∪ b)
2 or0 102 . . . 4 (b ∪ 0) = b
32ax-r1 35 . . 3 b = (b ∪ 0)
4 skr0.1 . . . . . 6 a = 1
54ax-r4 37 . . . . 5 a = 1
6 df-f 42 . . . . . 6 0 = 1
76ax-r1 35 . . . . 5 1 = 0
85, 7ax-r2 36 . . . 4 a = 0
98ax-r5 38 . . 3 (ab) = (0 ∪ b)
101, 3, 93tr1 63 . 2 b = (ab)
11 skr0.2 . 2 (ab) = 1
1210, 11ax-r2 36 1 b = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6  1wt 8  0wf 9 This theorem was proved from axioms:  ax-a2 31  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38 This theorem depends on definitions:  df-t 41  df-f 42 This theorem is referenced by:  skmp3  245  wql2lem3  290  wql2lem4  291  id5id0  352  wr5-2v  366  lem3.3.2  1046  wdwom  1104
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