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Theorem 3vded3 819
Description: A 3-variable theorem. Experiment with weak deduction theorem.
Hypotheses
Ref Expression
3vded3.1 (c ->0 C (a, c)) = 1
3vded3.2 (c ->0 a) = 1
3vded3.3 (c ->0 (a ->0 b)) = 1
Assertion
Ref Expression
3vded3 (c ->0 b) = 1
Proof of Theorem 3vded3
StepHypRef Expression
1 df-i0 43 . 2 (c ->0 b) = (c' v b)
2 ax-a3 32 . . 3 ((c' v a') v b) = (c' v (a' v b))
3 cmtrcom 190 . . . . . . . . . . . . . . . 16 C (c, a) = C (a, c)
43lor 70 . . . . . . . . . . . . . . 15 (c' v C (c, a)) = (c' v C (a, c))
5 df-i0 43 . . . . . . . . . . . . . . 15 (c ->0 C (c, a)) = (c' v C (c, a))
6 df-i0 43 . . . . . . . . . . . . . . 15 (c ->0 C (a, c)) = (c' v C (a, c))
74, 5, 63tr1 63 . . . . . . . . . . . . . 14 (c ->0 C (c, a)) = (c ->0 C (a, c))
8 3vded3.1 . . . . . . . . . . . . . 14 (c ->0 C (a, c)) = 1
97, 8ax-r2 36 . . . . . . . . . . . . 13 (c ->0 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' ^ (c' v a)) = ((a' ^ c') v (a' ^ a))
16 df-i0 43 . . . . . . . . . . . 12 (c ->0 a) = (c' v a)
1716ax-r1 35 . . . . . . . . . . 11 (c' v a) = (c ->0 a)
18 3vded3.2 . . . . . . . . . . 11 (c ->0 a) = 1
1917, 18ax-r2 36 . . . . . . . . . 10 (c' v a) = 1
2019lan 77 . . . . . . . . 9 (a' ^ (c' v a)) = (a' ^ 1)
21 dff 101 . . . . . . . . . . . . 13 0 = (a ^ a')
22 ancom 74 . . . . . . . . . . . . 13 (a ^ a') = (a' ^ a)
2321, 22ax-r2 36 . . . . . . . . . . . 12 0 = (a' ^ a)
2423lor 70 . . . . . . . . . . 11 ((a' ^ c') v 0) = ((a' ^ c') v (a' ^ a))
2524ax-r1 35 . . . . . . . . . 10 ((a' ^ c') v (a' ^ a)) = ((a' ^ c') v 0)
26 or0 102 . . . . . . . . . 10 ((a' ^ c') v 0) = (a' ^ c')
2725, 26ax-r2 36 . . . . . . . . 9 ((a' ^ c') v (a' ^ a)) = (a' ^ c')
2815, 20, 273tr2 64 . . . . . . . 8 (a' ^ 1) = (a' ^ c')
29 an1 106 . . . . . . . 8 (a' ^ 1) = a'
30 ancom 74 . . . . . . . 8 (a' ^ c') = (c' ^ a')
3128, 29, 303tr2 64 . . . . . . 7 a' = (c' ^ a')
3231lor 70 . . . . . 6 (c' v a') = (c' v (c' ^ a'))
33 orabs 120 . . . . . 6 (c' v (c' ^ a')) = c'
3432, 33ax-r2 36 . . . . 5 (c' v a') = c'
3534ax-r5 38 . . . 4 ((c' v a') v b) = (c' v b)
3635ax-r1 35 . . 3 (c' v b) = ((c' v a') v b)
37 df-i0 43 . . . 4 (c ->0 (a ->0 b)) = (c' v (a ->0 b))
38 df-i0 43 . . . . 5 (a ->0 b) = (a' v b)
3938lor 70 . . . 4 (c' v (a ->0 b)) = (c' v (a' v b))
4037, 39ax-r2 36 . . 3 (c ->0 (a ->0 b)) = (c' v (a' v b))
412, 36, 403tr1 63 . 2 (c' v b) = (c ->0 (a ->0 b))
42 3vded3.3 . 2 (c ->0 (a ->0 b)) = 1
431, 41, 423tr 65 1 (c ->0 b) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v 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)
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