Theorem oadist12 990

Description: Distributive law derived from OA.

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 753	. 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 989	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)))))

Syntax hints:   = wb 1   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₂ wi2 14

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-3oa 978

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i0 42  df-i1 43  df-i2 44  df-le1 122  df-le2 123

metamath.org