Theorem oa43v 1028

Description: Derivation of 3-variable OA from 4-variable OA. (Contributed by NM, 28-Nov-1998.)

Assertion

Ref	Expression
oa43v	((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)

Proof of Theorem oa43v

Step	Hyp	Ref	Expression
1		ud2lem0c 278	. . . . 5 (a →2 c)⊥ = (c⊥ ∩ (a ∪ c))
2		lea 160	. . . . 5 (c⊥ ∩ (a ∪ c)) ≤ c⊥
3	1, 2	bltr 138	. . . 4 (a →2 c)⊥ ≤ c⊥
4		ud2lem0c 278	. . . . 5 (a →2 b)⊥ = (b⊥ ∩ (a ∪ b))
5		lea 160	. . . . 5 (b⊥ ∩ (a ∪ b)) ≤ b⊥
6	4, 5	bltr 138	. . . 4 (a →2 b)⊥ ≤ b⊥
7	3, 6	ax-oal4 1026	. . . 4 (((a →2 c)⊥ ∪ c) ∩ ((a →2 b)⊥ ∪ b)) ≤ (c ∪ ((a →2 c)⊥ ∩ ((a →2 b)⊥ ∪ (((a →2 c)⊥ ∪ (a →2 b)⊥ ) ∩ (c ∪ b)))))
8		id 59	. . . 4 (a →2 c)⊥ = (a →2 c)⊥
9		id 59	. . . 4 (a →2 b)⊥ = (a →2 b)⊥
10	3, 6, 7, 8, 9	oa4v3v 934	. . 3 (c⊥ ∩ ((a →2 c) ∪ ((a →2 b) ∩ ((c ∪ b)⊥ ∪ ((a →2 c) ∩ (a →2 b)))))) ≤ ((c⊥ ∩ (a →2 c)) ∪ (b⊥ ∩ (a →2 b)))
11	10	oal42 935	. 2 (c⊥ ∩ ((a →2 c) ∪ ((a →2 b) ∩ ((c ∪ b)⊥ ∪ ((a →2 c) ∩ (a →2 b)))))) ≤ a⊥
12	11	oa23 936	1 ((a →2 b) ∩ ((b ∪ c)⊥ ∪ ((a →2 b) ∩ (a →2 c)))) ≤ (a →2 c)

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-oal4 1026

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)