Theorem oadist2b 1008

Description: Distributive inference derived from OA. (Contributed by NM, 17-Nov-1998.)

Hypothesis

Ref	Expression
oadist2b.1	d ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))

Assertion

Ref	Expression
oadist2b	((a →2 b) ∩ (d ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))) = (((a →2 b) ∩ d) ∪ ((a →2 b) ∩ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))))

Proof of Theorem oadist2b

Step	Hyp	Ref	Expression
1		oadist2b.1	. . . . 5 d ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
2		u12lem 771	. . . . . 6 (((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))) = ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
3	2	ax-r1 35	. . . . 5 ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c))) = (((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))
4	1, 3	lbtr 139	. . . 4 d ≤ (((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))
5		leor 159	. . . 4 ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))) ≤ (((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))
6	4, 5	lel2or 170	. . 3 (d ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))) ≤ (((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))
7	6, 2	lbtr 139	. 2 (d ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))) ≤ ((b ∪ c) →0 ((a →2 b) ∩ (a →2 c)))
8	7	oadist2a 1007	1 ((a →2 b) ∩ (d ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))) = (((a →2 b) ∩ d) ∪ ((a →2 b) ∩ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))))

Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₁ wi1 12   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-3oa 998

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i0 43  df-i1 44  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by: (None)