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Theorem 3vth6 809
 Description: A 3-variable theorem.
Assertion
Ref Expression
3vth6 ((a2 b)2 (bc)) = (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))

Proof of Theorem 3vth6
StepHypRef Expression
1 oridm 110 . . 3 (((a2 b)2 (bc)) ∪ ((a2 b)2 (bc))) = ((a2 b)2 (bc))
21ax-r1 35 . 2 ((a2 b)2 (bc)) = (((a2 b)2 (bc)) ∪ ((a2 b)2 (bc)))
3 3vth4 807 . . . 4 ((a2 b)2 (bc)) = ((a2 c)2 (bc))
43lor 70 . . 3 (((a2 b)2 (bc)) ∪ ((a2 b)2 (bc))) = (((a2 b)2 (bc)) ∪ ((a2 c)2 (bc)))
5 3vth5 808 . . . . 5 ((a2 b)2 (bc)) = (c ∪ ((a2 b) ∩ (c2 b)))
6 ax-a2 31 . . . . . . 7 (bc) = (cb)
76ud2lem0a 258 . . . . . 6 ((a2 c)2 (bc)) = ((a2 c)2 (cb))
8 3vth5 808 . . . . . 6 ((a2 c)2 (cb)) = (b ∪ ((a2 c) ∩ (b2 c)))
97, 8ax-r2 36 . . . . 5 ((a2 c)2 (bc)) = (b ∪ ((a2 c) ∩ (b2 c)))
105, 92or 72 . . . 4 (((a2 b)2 (bc)) ∪ ((a2 c)2 (bc))) = ((c ∪ ((a2 b) ∩ (c2 b))) ∪ (b ∪ ((a2 c) ∩ (b2 c))))
11 or4 84 . . . . 5 ((c ∪ ((a2 b) ∩ (c2 b))) ∪ (b ∪ ((a2 c) ∩ (b2 c)))) = ((cb) ∪ (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c))))
12 ax-a2 31 . . . . . . 7 (cb) = (bc)
1312ax-r5 38 . . . . . 6 ((cb) ∪ (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))) = ((bc) ∪ (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c))))
14 or4 84 . . . . . . 7 ((bc) ∪ (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))) = ((b ∪ ((a2 b) ∩ (c2 b))) ∪ (c ∪ ((a2 c) ∩ (b2 c))))
15 leo 158 . . . . . . . . . . 11 b ≤ (b ∪ (ab ))
16 df-i2 45 . . . . . . . . . . . 12 (a2 b) = (b ∪ (ab ))
1716ax-r1 35 . . . . . . . . . . 11 (b ∪ (ab )) = (a2 b)
1815, 17lbtr 139 . . . . . . . . . 10 b ≤ (a2 b)
19 leo 158 . . . . . . . . . . 11 b ≤ (b ∪ (cb ))
20 df-i2 45 . . . . . . . . . . . 12 (c2 b) = (b ∪ (cb ))
2120ax-r1 35 . . . . . . . . . . 11 (b ∪ (cb )) = (c2 b)
2219, 21lbtr 139 . . . . . . . . . 10 b ≤ (c2 b)
2318, 22ler2an 173 . . . . . . . . 9 b ≤ ((a2 b) ∩ (c2 b))
2423df-le2 131 . . . . . . . 8 (b ∪ ((a2 b) ∩ (c2 b))) = ((a2 b) ∩ (c2 b))
25 leo 158 . . . . . . . . . . 11 c ≤ (c ∪ (ac ))
26 df-i2 45 . . . . . . . . . . . 12 (a2 c) = (c ∪ (ac ))
2726ax-r1 35 . . . . . . . . . . 11 (c ∪ (ac )) = (a2 c)
2825, 27lbtr 139 . . . . . . . . . 10 c ≤ (a2 c)
29 leo 158 . . . . . . . . . . 11 c ≤ (c ∪ (bc ))
30 df-i2 45 . . . . . . . . . . . 12 (b2 c) = (c ∪ (bc ))
3130ax-r1 35 . . . . . . . . . . 11 (c ∪ (bc )) = (b2 c)
3229, 31lbtr 139 . . . . . . . . . 10 c ≤ (b2 c)
3328, 32ler2an 173 . . . . . . . . 9 c ≤ ((a2 c) ∩ (b2 c))
3433df-le2 131 . . . . . . . 8 (c ∪ ((a2 c) ∩ (b2 c))) = ((a2 c) ∩ (b2 c))
3524, 342or 72 . . . . . . 7 ((b ∪ ((a2 b) ∩ (c2 b))) ∪ (c ∪ ((a2 c) ∩ (b2 c)))) = (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))
3614, 35ax-r2 36 . . . . . 6 ((bc) ∪ (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))) = (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))
3713, 36ax-r2 36 . . . . 5 ((cb) ∪ (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))) = (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))
3811, 37ax-r2 36 . . . 4 ((c ∪ ((a2 b) ∩ (c2 b))) ∪ (b ∪ ((a2 c) ∩ (b2 c)))) = (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))
3910, 38ax-r2 36 . . 3 (((a2 b)2 (bc)) ∪ ((a2 c)2 (bc))) = (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))
404, 39ax-r2 36 . 2 (((a2 b)2 (bc)) ∪ ((a2 b)2 (bc))) = (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))
412, 40ax-r2 36 1 ((a2 b)2 (bc)) = (((a2 b) ∩ (c2 b)) ∪ ((a2 c) ∩ (b2 c)))
 Colors of variables: term Syntax hints:   = wb 1  ⊥ wn 4   ∪ 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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