Theorem 3vth3 806

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

Assertion

Ref	Expression
3vth3	((a →2 c) ∪ ((a →2 b) ∩ (b ∪ c)⊥ )) = (a →2 c)

Proof of Theorem 3vth3

Step	Hyp	Ref	Expression
1		ax-a2 31	. 2 ((a →2 c) ∪ ((a →2 b) ∩ (b ∪ c)⊥ )) = (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ (a →2 c))
2		3vth1 804	. . 3 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ (a →2 c)
3	2	df-le2 131	. 2 (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ (a →2 c)) = (a →2 c)
4	1, 3	ax-r2 36	1 ((a →2 c) ∪ ((a →2 b) ∩ (b ∪ c)⊥ )) = (a →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-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

This theorem is referenced by: (None)