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Mirrors > Home > QLE Home > Th. List > ska4 | Unicode version |
Description: Soundness theorem for Kalmbach's quantum propositional logic axiom KA4. |
Ref | Expression |
---|---|
ska4 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfnb 95 | . . 3 | |
2 | dfb 94 | . . 3 | |
3 | 1, 2 | 2or 72 | . 2 |
4 | ax-a2 31 | . 2 | |
5 | ax-a3 32 | . . 3 | |
6 | le1 146 | . . . . . . . . 9 | |
7 | df-t 41 | . . . . . . . . . . 11 | |
8 | oran 87 | . . . . . . . . . . . . 13 | |
9 | 8 | lor 70 | . . . . . . . . . . . 12 |
10 | 9 | ax-r1 35 | . . . . . . . . . . 11 |
11 | 7, 10 | ax-r2 36 | . . . . . . . . . 10 |
12 | lea 160 | . . . . . . . . . . . . 13 | |
13 | 12 | lecon 154 | . . . . . . . . . . . 12 |
14 | lea 160 | . . . . . . . . . . . . 13 | |
15 | 14 | lecon 154 | . . . . . . . . . . . 12 |
16 | 13, 15 | le2an 169 | . . . . . . . . . . 11 |
17 | 16 | leror 152 | . . . . . . . . . 10 |
18 | 11, 17 | bltr 138 | . . . . . . . . 9 |
19 | 6, 18 | lebi 145 | . . . . . . . 8 |
20 | 19 | ran 78 | . . . . . . 7 |
21 | ancom 74 | . . . . . . 7 | |
22 | an1 106 | . . . . . . 7 | |
23 | 20, 21, 22 | 3tr 65 | . . . . . 6 |
24 | 23 | lor 70 | . . . . 5 |
25 | le1 146 | . . . . . 6 | |
26 | df-t 41 | . . . . . . . 8 | |
27 | anandir 115 | . . . . . . . . 9 | |
28 | oran3 93 | . . . . . . . . . . . . 13 | |
29 | 28 | ax-r5 38 | . . . . . . . . . . . 12 |
30 | oran3 93 | . . . . . . . . . . . 12 | |
31 | 29, 30 | ax-r2 36 | . . . . . . . . . . 11 |
32 | 31 | ax-r1 35 | . . . . . . . . . 10 |
33 | ax-a2 31 | . . . . . . . . . 10 | |
34 | 32, 33 | ax-r2 36 | . . . . . . . . 9 |
35 | 27, 34 | 2or 72 | . . . . . . . 8 |
36 | 26, 35 | ax-r2 36 | . . . . . . 7 |
37 | lear 161 | . . . . . . . . . . 11 | |
38 | 37 | lecon 154 | . . . . . . . . . 10 |
39 | lear 161 | . . . . . . . . . . 11 | |
40 | 39 | lecon 154 | . . . . . . . . . 10 |
41 | 38, 40 | ler2an 173 | . . . . . . . . 9 |
42 | 41 | leror 152 | . . . . . . . 8 |
43 | 42 | lelor 166 | . . . . . . 7 |
44 | 36, 43 | bltr 138 | . . . . . 6 |
45 | 25, 44 | lebi 145 | . . . . 5 |
46 | 24, 45 | ax-r2 36 | . . . 4 |
47 | wlea 388 | . . . . . . . . . . 11 | |
48 | wleo 387 | . . . . . . . . . . 11 | |
49 | 47, 48 | wletr 396 | . . . . . . . . . 10 |
50 | 49 | wlecom 409 | . . . . . . . . 9 |
51 | 50 | wcomcom 414 | . . . . . . . 8 |
52 | 51 | wcomcom2 415 | . . . . . . 7 |
53 | wlea 388 | . . . . . . . . . . 11 | |
54 | wleo 387 | . . . . . . . . . . . 12 | |
55 | ax-a2 31 | . . . . . . . . . . . . 13 | |
56 | 55 | bi1 118 | . . . . . . . . . . . 12 |
57 | 54, 56 | wlbtr 398 | . . . . . . . . . . 11 |
58 | 53, 57 | wletr 396 | . . . . . . . . . 10 |
59 | 58 | wlecom 409 | . . . . . . . . 9 |
60 | 59 | wcomcom 414 | . . . . . . . 8 |
61 | 60 | wcomcom2 415 | . . . . . . 7 |
62 | 52, 61 | wcom2an 428 | . . . . . 6 |
63 | wcomorr 412 | . . . . . . . . 9 | |
64 | 63 | wcomcom 414 | . . . . . . . 8 |
65 | 64 | wcomcom2 415 | . . . . . . 7 |
66 | wcomorr 412 | . . . . . . . . . 10 | |
67 | 66, 56 | wcbtr 411 | . . . . . . . . 9 |
68 | 67 | wcomcom 414 | . . . . . . . 8 |
69 | 68 | wcomcom2 415 | . . . . . . 7 |
70 | 65, 69 | wcom2or 427 | . . . . . 6 |
71 | 62, 70 | wfh4 426 | . . . . 5 |
72 | 71 | wlor 368 | . . . 4 |
73 | 46, 72 | wwbmpr 206 | . . 3 |
74 | 5, 73 | ax-r2 36 | . 2 |
75 | 3, 4, 74 | 3tr 65 | 1 |
Colors of variables: term |
Syntax hints: wb 1 wn 4 tb 5 wo 6 wa 7 wt 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 ax-wom 361 |
This theorem depends on definitions: df-b 39 df-a 40 df-t 41 df-f 42 df-i1 44 df-i2 45 df-le 129 df-le1 130 df-le2 131 df-cmtr 134 |
This theorem is referenced by: wom2 434 u3lemax4 796 |
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