Theorem axoa4a 1037

Description: Proper 4-variable OA law variant. (Contributed by NM, 22-Dec-1998.)

Assertion

Ref	Expression
axoa4a	((a →1 d) ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))))) ≤ (((a ∩ d) ∪ (b ∩ d)) ∪ (c ∩ d))

Proof of Theorem axoa4a

Step	Hyp	Ref	Expression
1		id 59	. 2 (a →1 d)⊥ = (a →1 d)⊥
2		id 59	. 2 (b →1 d)⊥ = (b →1 d)⊥
3		id 59	. 2 (c →1 d)⊥ = (c →1 d)⊥
4		leo 158	. . . 4 a⊥ ≤ (a⊥ ∪ (a ∩ d))
5		df-i1 44	. . . . . 6 (a →1 d) = (a⊥ ∪ (a ∩ d))
6	5	ax-r1 35	. . . . 5 (a⊥ ∪ (a ∩ d)) = (a →1 d)
7		ax-a1 30	. . . . 5 (a →1 d) = (a →1 d)⊥ ⊥
8	6, 7	ax-r2 36	. . . 4 (a⊥ ∪ (a ∩ d)) = (a →1 d)⊥ ⊥
9	4, 8	lbtr 139	. . 3 a⊥ ≤ (a →1 d)⊥ ⊥
10		leo 158	. . . 4 b⊥ ≤ (b⊥ ∪ (b ∩ d))
11		df-i1 44	. . . . . 6 (b →1 d) = (b⊥ ∪ (b ∩ d))
12	11	ax-r1 35	. . . . 5 (b⊥ ∪ (b ∩ d)) = (b →1 d)
13		ax-a1 30	. . . . 5 (b →1 d) = (b →1 d)⊥ ⊥
14	12, 13	ax-r2 36	. . . 4 (b⊥ ∪ (b ∩ d)) = (b →1 d)⊥ ⊥
15	10, 14	lbtr 139	. . 3 b⊥ ≤ (b →1 d)⊥ ⊥
16		leo 158	. . . 4 c⊥ ≤ (c⊥ ∪ (c ∩ d))
17		df-i1 44	. . . . . 6 (c →1 d) = (c⊥ ∪ (c ∩ d))
18	17	ax-r1 35	. . . . 5 (c⊥ ∪ (c ∩ d)) = (c →1 d)
19		ax-a1 30	. . . . 5 (c →1 d) = (c →1 d)⊥ ⊥
20	18, 19	ax-r2 36	. . . 4 (c⊥ ∪ (c ∩ d)) = (c →1 d)⊥ ⊥
21	16, 20	lbtr 139	. . 3 c⊥ ≤ (c →1 d)⊥ ⊥
22	9, 15, 21	oa6 1036	. 2 (((a⊥ ∪ (a →1 d)⊥ ) ∩ (b⊥ ∪ (b →1 d)⊥ )) ∩ (c⊥ ∪ (c →1 d)⊥ )) ≤ ((a →1 d)⊥ ∪ (a⊥ ∩ (b⊥ ∪ (((a⊥ ∪ b⊥ ) ∩ ((a →1 d)⊥ ∪ (b →1 d)⊥ )) ∩ (((a⊥ ∪ c⊥ ) ∩ ((a →1 d)⊥ ∪ (c →1 d)⊥ )) ∪ ((b⊥ ∪ c⊥ ) ∩ ((b →1 d)⊥ ∪ (c →1 d)⊥ )))))))
23	1, 2, 3, 22	oa6to4 958	1 ((a →1 d) ∩ (a ∪ (b ∩ (((a ∩ b) ∪ ((a →1 d) ∩ (b →1 d))) ∪ (((a ∩ c) ∪ ((a →1 d) ∩ (c →1 d))) ∩ ((b ∩ c) ∪ ((b →1 d) ∩ (c →1 d)))))))) ≤ (((a ∩ d) ∪ (b ∩ d)) ∪ (c ∩ d))

This proof depends on syntax axioms:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 12

This proof depends on 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-4oa 1033

This proof depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is used by:  4oa  1039