Theorem oadist12 1010

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

Assertion

Ref	Expression
oadist12	((a →2 b) ∩ (((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))) = (((a →2 b) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c)))) ∪ ((a →2 b) ∩ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))))

Proof of Theorem oadist12

Step	Hyp	Ref	Expression
1		u12lem 771	. 2 (((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)))
2	1	oadist2 1009	1 ((a →2 b) ∩ (((b ∪ c) →1 ((a →2 b) ∩ (a →2 c))) ∪ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c))))) = (((a →2 b) ∩ ((b ∪ c) →1 ((a →2 b) ∩ (a →2 c)))) ∪ ((a →2 b) ∩ ((b ∪ c) →2 ((a →2 b) ∩ (a →2 c)))))

This proof depends on syntax axioms:   = wb 1   ∪ wo 6   ∩ wa 7   →₁ wi1 12   →₂ wi2 13

This proof depends on 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 proof 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 used by: (None)