Theorem oa64v 1031

Description: Derivation of 4-variable OA from 6-variable OA. (Contributed by NM, 29-Nov-1998.)

Hypotheses

Ref	Expression
oa64v.1	a ≤ b⊥
oa64v.2	c ≤ d⊥

Assertion

Ref	Expression
oa64v	((a ∪ b) ∩ (c ∪ d)) ≤ (b ∪ (a ∩ (c ∪ ((a ∪ c) ∩ (b ∪ d)))))

Proof of Theorem oa64v

Step	Hyp	Ref	Expression
1		oa64v.1	. . 3 a ≤ b⊥
2		oa64v.2	. . 3 c ≤ d⊥
3		le0 147	. . 3 0 ≤ 1⊥
4	1, 2, 3	ax-oa6 1030	. 2 (((a ∪ b) ∩ (c ∪ d)) ∩ (0 ∪ 1)) ≤ (b ∪ (a ∩ (c ∪ (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ 0) ∩ (b ∪ 1)) ∪ ((c ∪ 0) ∩ (d ∪ 1)))))))
5		id 59	. 2 0 = 0
6		id 59	. 2 1 = 1
7	4, 5, 6	oa6v4v 933	1 ((a ∪ b) ∩ (c ∪ d)) ≤ (b ∪ (a ∩ (c ∪ ((a ∪ c) ∩ (b ∪ d)))))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8  0wf 9

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-oa6 1030

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131

This theorem is referenced by:  oa63v  1032