Theorem 3vth3 788

Description: A 3-variable theorem. Equivalent to OML.

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 30	. 2 ((a →2 c) ∪ ((a →2 b) ∩ (b ∪ c)⊥ )) = (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ (a →2 c))
2		3vth1 786	. . 3 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ (a →2 c)
3	2	df-le2 123	. 2 (((a →2 b) ∩ (b ∪ c)⊥ ) ∪ (a →2 c)) = (a →2 c)
4	1, 3	ax-r2 35	1 ((a →2 c) ∪ ((a →2 b) ∩ (b ∪ c)⊥ )) = (a →2 c)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →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-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123

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