Theorem oa64v 1010

Description: Derivation of 4-variable OA from 6-variable OA.

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 139	. . 3 0 ≤ 1⊥
4	1, 2, 3	ax-oa6 1009	. 2 (((a ∪ b) ∩ (c ∪ d)) ∩ (0 ∪ 1)) ≤ (b ∪ (a ∩ (c ∪ (((a ∪ c) ∩ (b ∪ d)) ∩ (((a ∪ 0) ∩ (b ∪ 1)) ∪ ((c ∪ 0) ∩ (d ∪ 1)))))))
5		id 58	. 2 0 = 0
6		id 58	. 2 1 = 1
7	4, 5, 6	oa6v4v 913	1 ((a ∪ b) ∩ (c ∪ d)) ≤ (b ∪ (a ∩ (c ∪ ((a ∪ c) ∩ (b ∪ d)))))

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9  0wf 10

This theorem is referenced by:  oa63v 1011

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-oa6 1009

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123

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