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Theorem 3vth6 809
Description: A 3-variable theorem.
Assertion
Ref Expression
3vth6 ((a ->2 b)' ->2 (b v c)) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
Proof of Theorem 3vth6
StepHypRef Expression
1 oridm 110 . . 3 (((a ->2 b)' ->2 (b v c)) v ((a ->2 b)' ->2 (b v c))) = ((a ->2 b)' ->2 (b v c))
21ax-r1 35 . 2 ((a ->2 b)' ->2 (b v c)) = (((a ->2 b)' ->2 (b v c)) v ((a ->2 b)' ->2 (b v c)))
3 3vth4 807 . . . 4 ((a ->2 b)' ->2 (b v c)) = ((a ->2 c)' ->2 (b v c))
43lor 70 . . 3 (((a ->2 b)' ->2 (b v c)) v ((a ->2 b)' ->2 (b v c))) = (((a ->2 b)' ->2 (b v c)) v ((a ->2 c)' ->2 (b v c)))
5 3vth5 808 . . . . 5 ((a ->2 b)' ->2 (b v c)) = (c v ((a ->2 b) ^ (c ->2 b)))
6 ax-a2 31 . . . . . . 7 (b v c) = (c v b)
76ud2lem0a 258 . . . . . 6 ((a ->2 c)' ->2 (b v c)) = ((a ->2 c)' ->2 (c v b))
8 3vth5 808 . . . . . 6 ((a ->2 c)' ->2 (c v b)) = (b v ((a ->2 c) ^ (b ->2 c)))
97, 8ax-r2 36 . . . . 5 ((a ->2 c)' ->2 (b v c)) = (b v ((a ->2 c) ^ (b ->2 c)))
105, 92or 72 . . . 4 (((a ->2 b)' ->2 (b v c)) v ((a ->2 c)' ->2 (b v c))) = ((c v ((a ->2 b) ^ (c ->2 b))) v (b v ((a ->2 c) ^ (b ->2 c))))
11 or4 84 . . . . 5 ((c v ((a ->2 b) ^ (c ->2 b))) v (b v ((a ->2 c) ^ (b ->2 c)))) = ((c v b) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c))))
12 ax-a2 31 . . . . . . 7 (c v b) = (b v c)
1312ax-r5 38 . . . . . 6 ((c v b) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))) = ((b v c) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c))))
14 or4 84 . . . . . . 7 ((b v c) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))) = ((b v ((a ->2 b) ^ (c ->2 b))) v (c v ((a ->2 c) ^ (b ->2 c))))
15 leo 158 . . . . . . . . . . 11 b =< (b v (a' ^ b'))
16 df-i2 45 . . . . . . . . . . . 12 (a ->2 b) = (b v (a' ^ b'))
1716ax-r1 35 . . . . . . . . . . 11 (b v (a' ^ b')) = (a ->2 b)
1815, 17lbtr 139 . . . . . . . . . 10 b =< (a ->2 b)
19 leo 158 . . . . . . . . . . 11 b =< (b v (c' ^ b'))
20 df-i2 45 . . . . . . . . . . . 12 (c ->2 b) = (b v (c' ^ b'))
2120ax-r1 35 . . . . . . . . . . 11 (b v (c' ^ b')) = (c ->2 b)
2219, 21lbtr 139 . . . . . . . . . 10 b =< (c ->2 b)
2318, 22ler2an 173 . . . . . . . . 9 b =< ((a ->2 b) ^ (c ->2 b))
2423df-le2 131 . . . . . . . 8 (b v ((a ->2 b) ^ (c ->2 b))) = ((a ->2 b) ^ (c ->2 b))
25 leo 158 . . . . . . . . . . 11 c =< (c v (a' ^ c'))
26 df-i2 45 . . . . . . . . . . . 12 (a ->2 c) = (c v (a' ^ c'))
2726ax-r1 35 . . . . . . . . . . 11 (c v (a' ^ c')) = (a ->2 c)
2825, 27lbtr 139 . . . . . . . . . 10 c =< (a ->2 c)
29 leo 158 . . . . . . . . . . 11 c =< (c v (b' ^ c'))
30 df-i2 45 . . . . . . . . . . . 12 (b ->2 c) = (c v (b' ^ c'))
3130ax-r1 35 . . . . . . . . . . 11 (c v (b' ^ c')) = (b ->2 c)
3229, 31lbtr 139 . . . . . . . . . 10 c =< (b ->2 c)
3328, 32ler2an 173 . . . . . . . . 9 c =< ((a ->2 c) ^ (b ->2 c))
3433df-le2 131 . . . . . . . 8 (c v ((a ->2 c) ^ (b ->2 c))) = ((a ->2 c) ^ (b ->2 c))
3524, 342or 72 . . . . . . 7 ((b v ((a ->2 b) ^ (c ->2 b))) v (c v ((a ->2 c) ^ (b ->2 c)))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
3614, 35ax-r2 36 . . . . . 6 ((b v c) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
3713, 36ax-r2 36 . . . . 5 ((c v b) v (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
3811, 37ax-r2 36 . . . 4 ((c v ((a ->2 b) ^ (c ->2 b))) v (b v ((a ->2 c) ^ (b ->2 c)))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
3910, 38ax-r2 36 . . 3 (((a ->2 b)' ->2 (b v c)) v ((a ->2 c)' ->2 (b v c))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
404, 39ax-r2 36 . 2 (((a ->2 b)' ->2 (b v c)) v ((a ->2 b)' ->2 (b v c))) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
412, 40ax-r2 36 1 ((a ->2 b)' ->2 (b v c)) = (((a ->2 b) ^ (c ->2 b)) v ((a ->2 c) ^ (b ->2 c)))
Colors of variables: term
Syntax hints:   = wb 1  'wn 4   v wo 6   ^ wa 7   ->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:  3vth8  811
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