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Theorem oa63v 1032
 Description: Derivation of 3-variable OA from 6-variable OA.
Assertion
Ref Expression
oa63v ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)

Proof of Theorem oa63v
StepHypRef Expression
1 ud2lem0c 278 . . . . 5 (a2 c) = (c ∩ (ac))
2 lea 160 . . . . 5 (c ∩ (ac)) ≤ c
31, 2bltr 138 . . . 4 (a2 c)c
4 ud2lem0c 278 . . . . 5 (a2 b) = (b ∩ (ab))
5 lea 160 . . . . 5 (b ∩ (ab)) ≤ b
64, 5bltr 138 . . . 4 (a2 b)b
73, 6oa64v 1031 . . . 4 (((a2 c)c) ∩ ((a2 b)b)) ≤ (c ∪ ((a2 c) ∩ ((a2 b) ∪ (((a2 c) ∪ (a2 b) ) ∩ (cb)))))
8 id 59 . . . 4 (a2 c) = (a2 c)
9 id 59 . . . 4 (a2 b) = (a2 b)
103, 6, 7, 8, 9oa4v3v 934 . . 3 (c ∩ ((a2 c) ∪ ((a2 b) ∩ ((cb) ∪ ((a2 c) ∩ (a2 b)))))) ≤ ((c ∩ (a2 c)) ∪ (b ∩ (a2 b)))
1110oal42 935 . 2 (c ∩ ((a2 c) ∪ ((a2 b) ∩ ((cb) ∪ ((a2 c) ∩ (a2 b)))))) ≤ a
1211oa23 936 1 ((a2 b) ∩ ((bc) ∪ ((a2 b) ∩ (a2 c)))) ≤ (a2 c)
 Colors of variables: term Syntax hints:   ≤ wle 2  ⊥ 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  ax-oa6 1030 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: (None)
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