Theorem 3vth8 811

Description: A 3-variable theorem. (Contributed by NM, 18-Oct-1998.)

Assertion

Ref	Expression
3vth8	(a →2 (b ∪ c)) = (((a →2 b) ∩ (c →2 b)) ∪ ((a →2 c) ∩ (b →2 c)))

Proof of Theorem 3vth8

Step	Hyp	Ref	Expression
1		3vth7 810	. . 3 ((a →2 b)⊥ →2 (b ∪ c)) = (a →2 (b ∪ c))
2	1	ax-r1 35	. 2 (a →2 (b ∪ c)) = ((a →2 b)⊥ →2 (b ∪ c))
3		3vth6 809	. 2 ((a →2 b)⊥ →2 (b ∪ c)) = (((a →2 b) ∩ (c →2 b)) ∪ ((a →2 c) ∩ (b →2 c)))
4	2, 3	ax-r2 36	1 (a →2 (b ∪ c)) = (((a →2 b) ∩ (c →2 b)) ∪ ((a →2 c) ∩ (b →2 c)))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This theorem was proved from 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

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)