Quantum Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  QLE Home  >  Th. List  >  3vded3 GIF version

Theorem 3vded3 819
 Description: A 3-variable theorem. Experiment with weak deduction theorem.
Hypotheses
Ref Expression
3vded3.1 (c0 C (a, c)) = 1
3vded3.2 (c0 a) = 1
3vded3.3 (c0 (a0 b)) = 1
Assertion
Ref Expression
3vded3 (c0 b) = 1

Proof of Theorem 3vded3
StepHypRef Expression
1 df-i0 43 . 2 (c0 b) = (cb)
2 ax-a3 32 . . 3 ((ca ) ∪ b) = (c ∪ (ab))
3 cmtrcom 190 . . . . . . . . . . . . . . . 16 C (c, a) = C (a, c)
43lor 70 . . . . . . . . . . . . . . 15 (c C (c, a)) = (c C (a, c))
5 df-i0 43 . . . . . . . . . . . . . . 15 (c0 C (c, a)) = (c C (c, a))
6 df-i0 43 . . . . . . . . . . . . . . 15 (c0 C (a, c)) = (c C (a, c))
74, 5, 63tr1 63 . . . . . . . . . . . . . 14 (c0 C (c, a)) = (c0 C (a, c))
8 3vded3.1 . . . . . . . . . . . . . 14 (c0 C (a, c)) = 1
97, 8ax-r2 36 . . . . . . . . . . . . 13 (c0 C (c, a)) = 1
109i0cmtrcom 495 . . . . . . . . . . . 12 c C a
1110comcom4 455 . . . . . . . . . . 11 c C a
1211comcom 453 . . . . . . . . . 10 a C c
13 comid 187 . . . . . . . . . . 11 a C a
1413comcom3 454 . . . . . . . . . 10 a C a
1512, 14fh1 469 . . . . . . . . 9 (a ∩ (ca)) = ((ac ) ∪ (aa))
16 df-i0 43 . . . . . . . . . . . 12 (c0 a) = (ca)
1716ax-r1 35 . . . . . . . . . . 11 (ca) = (c0 a)
18 3vded3.2 . . . . . . . . . . 11 (c0 a) = 1
1917, 18ax-r2 36 . . . . . . . . . 10 (ca) = 1
2019lan 77 . . . . . . . . 9 (a ∩ (ca)) = (a ∩ 1)
21 dff 101 . . . . . . . . . . . . 13 0 = (aa )
22 ancom 74 . . . . . . . . . . . . 13 (aa ) = (aa)
2321, 22ax-r2 36 . . . . . . . . . . . 12 0 = (aa)
2423lor 70 . . . . . . . . . . 11 ((ac ) ∪ 0) = ((ac ) ∪ (aa))
2524ax-r1 35 . . . . . . . . . 10 ((ac ) ∪ (aa)) = ((ac ) ∪ 0)
26 or0 102 . . . . . . . . . 10 ((ac ) ∪ 0) = (ac )
2725, 26ax-r2 36 . . . . . . . . 9 ((ac ) ∪ (aa)) = (ac )
2815, 20, 273tr2 64 . . . . . . . 8 (a ∩ 1) = (ac )
29 an1 106 . . . . . . . 8 (a ∩ 1) = a
30 ancom 74 . . . . . . . 8 (ac ) = (ca )
3128, 29, 303tr2 64 . . . . . . 7 a = (ca )
3231lor 70 . . . . . 6 (ca ) = (c ∪ (ca ))
33 orabs 120 . . . . . 6 (c ∪ (ca )) = c
3432, 33ax-r2 36 . . . . 5 (ca ) = c
3534ax-r5 38 . . . 4 ((ca ) ∪ b) = (cb)
3635ax-r1 35 . . 3 (cb) = ((ca ) ∪ b)
37 df-i0 43 . . . 4 (c0 (a0 b)) = (c ∪ (a0 b))
38 df-i0 43 . . . . 5 (a0 b) = (ab)
3938lor 70 . . . 4 (c ∪ (a0 b)) = (c ∪ (ab))
4037, 39ax-r2 36 . . 3 (c0 (a0 b)) = (c ∪ (ab))
412, 36, 403tr1 63 . 2 (cb) = (c0 (a0 b))
42 3vded3.3 . 2 (c0 (a0 b)) = 1
431, 41, 423tr 65 1 (c0 b) = 1
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9   →0 wi0 11   C wcmtr 29 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-r3 439 This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i0 43  df-le1 130  df-le2 131  df-c1 132  df-c2 133  df-cmtr 134 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator