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Theorem 2oalem1 825
Description: Lemma for OA-like stuff with ->2 instead of ->0.
Assertion
Ref Expression
2oalem1 ((a ->2 b)' v ((b v c) v ((a ->2 b) ^ (a ->2 c)))) = 1
Proof of Theorem 2oalem1
StepHypRef Expression
1 or12 80 . 2 ((a ->2 b)' v ((b v c) v ((a ->2 b) ^ (a ->2 c)))) = ((b v c) v ((a ->2 b)' v ((a ->2 b) ^ (a ->2 c))))
2 ud2lem0c 278 . . . 4 (a ->2 b)' = (b' ^ (a v b))
3 df-i2 45 . . . . 5 (a ->2 b) = (b v (a' ^ b'))
4 df-i2 45 . . . . 5 (a ->2 c) = (c v (a' ^ c'))
53, 42an 79 . . . 4 ((a ->2 b) ^ (a ->2 c)) = ((b v (a' ^ b')) ^ (c v (a' ^ c')))
62, 52or 72 . . 3 ((a ->2 b)' v ((a ->2 b) ^ (a ->2 c))) = ((b' ^ (a v b)) v ((b v (a' ^ b')) ^ (c v (a' ^ c'))))
76lor 70 . 2 ((b v c) v ((a ->2 b)' v ((a ->2 b) ^ (a ->2 c)))) = ((b v c) v ((b' ^ (a v b)) v ((b v (a' ^ b')) ^ (c v (a' ^ c')))))
8 or32 82 . . . . 5 ((b v c) v (b' ^ (a v b))) = ((b v (b' ^ (a v b))) v c)
9 oml 445 . . . . . . 7 (b v (b' ^ (b v a))) = (b v a)
10 ax-a2 31 . . . . . . . . 9 (a v b) = (b v a)
1110lan 77 . . . . . . . 8 (b' ^ (a v b)) = (b' ^ (b v a))
1211lor 70 . . . . . . 7 (b v (b' ^ (a v b))) = (b v (b' ^ (b v a)))
139, 12, 103tr1 63 . . . . . 6 (b v (b' ^ (a v b))) = (a v b)
1413ax-r5 38 . . . . 5 ((b v (b' ^ (a v b))) v c) = ((a v b) v c)
158, 14ax-r2 36 . . . 4 ((b v c) v (b' ^ (a v b))) = ((a v b) v c)
1615ax-r5 38 . . 3 (((b v c) v (b' ^ (a v b))) v ((b v (a' ^ b')) ^ (c v (a' ^ c')))) = (((a v b) v c) v ((b v (a' ^ b')) ^ (c v (a' ^ c'))))
17 ax-a3 32 . . 3 (((b v c) v (b' ^ (a v b))) v ((b v (a' ^ b')) ^ (c v (a' ^ c')))) = ((b v c) v ((b' ^ (a v b)) v ((b v (a' ^ b')) ^ (c v (a' ^ c')))))
18 oran 87 . . . . . . . . . . . . 13 (a v b) = (a' ^ b')'
1918lan 77 . . . . . . . . . . . 12 (b' ^ (a v b)) = (b' ^ (a' ^ b')')
20 anor3 90 . . . . . . . . . . . 12 (b' ^ (a' ^ b')') = (b v (a' ^ b'))'
2119, 20ax-r2 36 . . . . . . . . . . 11 (b' ^ (a v b)) = (b v (a' ^ b'))'
2221ax-r1 35 . . . . . . . . . 10 (b v (a' ^ b'))' = (b' ^ (a v b))
23 lear 161 . . . . . . . . . 10 (b' ^ (a v b)) =< (a v b)
2422, 23bltr 138 . . . . . . . . 9 (b v (a' ^ b'))' =< (a v b)
25 leo 158 . . . . . . . . 9 (a v b) =< ((a v b) v c)
2624, 25letr 137 . . . . . . . 8 (b v (a' ^ b'))' =< ((a v b) v c)
2726lecom 180 . . . . . . 7 (b v (a' ^ b'))' C ((a v b) v c)
2827comcom6 459 . . . . . 6 (b v (a' ^ b')) C ((a v b) v c)
2928comcom 453 . . . . 5 ((a v b) v c) C (b v (a' ^ b'))
30 lear 161 . . . . . . . . . 10 (c' ^ (a v c)) =< (a v c)
31 leo 158 . . . . . . . . . 10 (a v c) =< ((a v c) v b)
3230, 31letr 137 . . . . . . . . 9 (c' ^ (a v c)) =< ((a v c) v b)
33 oran 87 . . . . . . . . . . 11 (a v c) = (a' ^ c')'
3433lan 77 . . . . . . . . . 10 (c' ^ (a v c)) = (c' ^ (a' ^ c')')
35 anor3 90 . . . . . . . . . 10 (c' ^ (a' ^ c')') = (c v (a' ^ c'))'
3634, 35ax-r2 36 . . . . . . . . 9 (c' ^ (a v c)) = (c v (a' ^ c'))'
37 or32 82 . . . . . . . . 9 ((a v c) v b) = ((a v b) v c)
3832, 36, 37le3tr2 141 . . . . . . . 8 (c v (a' ^ c'))' =< ((a v b) v c)
3938lecom 180 . . . . . . 7 (c v (a' ^ c'))' C ((a v b) v c)
4039comcom6 459 . . . . . 6 (c v (a' ^ c')) C ((a v b) v c)
4140comcom 453 . . . . 5 ((a v b) v c) C (c v (a' ^ c'))
4229, 41fh3 471 . . . 4 (((a v b) v c) v ((b v (a' ^ b')) ^ (c v (a' ^ c')))) = ((((a v b) v c) v (b v (a' ^ b'))) ^ (((a v b) v c) v (c v (a' ^ c'))))
43 or12 80 . . . . . 6 (((a v b) v c) v (b v (a' ^ b'))) = (b v (((a v b) v c) v (a' ^ b')))
44 or32 82 . . . . . . . 8 (((a v b) v c) v (a' ^ b')) = (((a v b) v (a' ^ b')) v c)
45 ax-a2 31 . . . . . . . 8 (((a v b) v (a' ^ b')) v c) = (c v ((a v b) v (a' ^ b')))
46 anor3 90 . . . . . . . . . . . 12 (a' ^ b') = (a v b)'
4746lor 70 . . . . . . . . . . 11 ((a v b) v (a' ^ b')) = ((a v b) v (a v b)')
48 df-t 41 . . . . . . . . . . . 12 1 = ((a v b) v (a v b)')
4948ax-r1 35 . . . . . . . . . . 11 ((a v b) v (a v b)') = 1
5047, 49ax-r2 36 . . . . . . . . . 10 ((a v b) v (a' ^ b')) = 1
5150lor 70 . . . . . . . . 9 (c v ((a v b) v (a' ^ b'))) = (c v 1)
52 or1 104 . . . . . . . . 9 (c v 1) = 1
5351, 52ax-r2 36 . . . . . . . 8 (c v ((a v b) v (a' ^ b'))) = 1
5444, 45, 533tr 65 . . . . . . 7 (((a v b) v c) v (a' ^ b')) = 1
5554lor 70 . . . . . 6 (b v (((a v b) v c) v (a' ^ b'))) = (b v 1)
56 or1 104 . . . . . 6 (b v 1) = 1
5743, 55, 563tr 65 . . . . 5 (((a v b) v c) v (b v (a' ^ b'))) = 1
58 or12 80 . . . . . 6 (((a v b) v c) v (c v (a' ^ c'))) = (c v (((a v b) v c) v (a' ^ c')))
59 or32 82 . . . . . . . . . 10 ((a v b) v c) = ((a v c) v b)
60 ax-a2 31 . . . . . . . . . 10 ((a v c) v b) = (b v (a v c))
6159, 60ax-r2 36 . . . . . . . . 9 ((a v b) v c) = (b v (a v c))
6261ax-r5 38 . . . . . . . 8 (((a v b) v c) v (a' ^ c')) = ((b v (a v c)) v (a' ^ c'))
63 ax-a3 32 . . . . . . . 8 ((b v (a v c)) v (a' ^ c')) = (b v ((a v c) v (a' ^ c')))
64 anor3 90 . . . . . . . . . . . 12 (a' ^ c') = (a v c)'
6564lor 70 . . . . . . . . . . 11 ((a v c) v (a' ^ c')) = ((a v c) v (a v c)')
66 df-t 41 . . . . . . . . . . . 12 1 = ((a v c) v (a v c)')
6766ax-r1 35 . . . . . . . . . . 11 ((a v c) v (a v c)') = 1
6865, 67ax-r2 36 . . . . . . . . . 10 ((a v c) v (a' ^ c')) = 1
6968lor 70 . . . . . . . . 9 (b v ((a v c) v (a' ^ c'))) = (b v 1)
7069, 56ax-r2 36 . . . . . . . 8 (b v ((a v c) v (a' ^ c'))) = 1
7162, 63, 703tr 65 . . . . . . 7 (((a v b) v c) v (a' ^ c')) = 1
7271lor 70 . . . . . 6 (c v (((a v b) v c) v (a' ^ c'))) = (c v 1)
7358, 72, 523tr 65 . . . . 5 (((a v b) v c) v (c v (a' ^ c'))) = 1
7457, 732an 79 . . . 4 ((((a v b) v c) v (b v (a' ^ b'))) ^ (((a v b) v c) v (c v (a' ^ c')))) = (1 ^ 1)
75 anidm 111 . . . 4 (1 ^ 1) = 1
7642, 74, 753tr 65 . . 3 (((a v b) v c) v ((b v (a' ^ b')) ^ (c v (a' ^ c')))) = 1
7716, 17, 763tr2 64 . 2 ((b v c) v ((b' ^ (a v b)) v ((b v (a' ^ b')) ^ (c v (a' ^ c'))))) = 1
781, 7, 773tr 65 1 ((a ->2 b)' v ((b v c) v ((a ->2 b) ^ (a ->2 c)))) = 1
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7  1wt 8   ->2 wi2 13
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-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133
This theorem is referenced by:  2oath1  826
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