Theorem 3vth2 805

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

Assertion

Ref	Expression
3vth2	((a →2 b) ∩ (b ∪ c)⊥ ) = ((a →2 c) ∩ (b ∪ c)⊥ )

Proof of Theorem 3vth2

Step	Hyp	Ref	Expression
1		3vth1 804	. . 3 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ (a →2 c)
2		lear 161	. . 3 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ (b ∪ c)⊥
3	1, 2	ler2an 173	. 2 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ ((a →2 c) ∩ (b ∪ c)⊥ )
4		ax-a2 31	. . . . . 6 (b ∪ c) = (c ∪ b)
5	4	ax-r4 37	. . . . 5 (b ∪ c)⊥ = (c ∪ b)⊥
6	5	lan 77	. . . 4 ((a →2 c) ∩ (b ∪ c)⊥ ) = ((a →2 c) ∩ (c ∪ b)⊥ )
7		3vth1 804	. . . 4 ((a →2 c) ∩ (c ∪ b)⊥ ) ≤ (a →2 b)
8	6, 7	bltr 138	. . 3 ((a →2 c) ∩ (b ∪ c)⊥ ) ≤ (a →2 b)
9		lear 161	. . 3 ((a →2 c) ∩ (b ∪ c)⊥ ) ≤ (b ∪ c)⊥
10	8, 9	ler2an 173	. 2 ((a →2 c) ∩ (b ∪ c)⊥ ) ≤ ((a →2 b) ∩ (b ∪ c)⊥ )
11	3, 10	lebi 145	1 ((a →2 b) ∩ (b ∪ c)⊥ ) = ((a →2 c) ∩ (b ∪ 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:  3vth4  807