Theorem 3vth1 804

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

Assertion

Ref	Expression
3vth1	((a →2 b) ∩ (b ∪ c)⊥ ) ≤ (a →2 c)

Proof of Theorem 3vth1

Step	Hyp	Ref	Expression
1		anor3 90	. . . . . . 7 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
2	1	lan 77	. . . . . 6 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∩ c⊥ )) = ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ )
3	2	ax-r1 35	. . . . 5 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ ) = ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∩ c⊥ ))
4		anass 76	. . . . . 6 (((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ∩ c⊥ ) = ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∩ c⊥ ))
5	4	ax-r1 35	. . . . 5 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∩ c⊥ )) = (((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ∩ c⊥ )
6	3, 5	ax-r2 36	. . . 4 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ ) = (((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ∩ c⊥ )
7		ancom 74	. . . . . . 7 ((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) = (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ )))
8		omlan 448	. . . . . . 7 (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ ))) = (b⊥ ∩ a⊥ )
9	7, 8	ax-r2 36	. . . . . 6 ((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) = (b⊥ ∩ a⊥ )
10		lear 161	. . . . . 6 (b⊥ ∩ a⊥ ) ≤ a⊥
11	9, 10	bltr 138	. . . . 5 ((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ≤ a⊥
12	11	leran 153	. . . 4 (((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ∩ c⊥ ) ≤ (a⊥ ∩ c⊥ )
13	6, 12	bltr 138	. . 3 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ ) ≤ (a⊥ ∩ c⊥ )
14		leor 159	. . 3 (a⊥ ∩ c⊥ ) ≤ (c ∪ (a⊥ ∩ c⊥ ))
15	13, 14	letr 137	. 2 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ ) ≤ (c ∪ (a⊥ ∩ c⊥ ))
16		df-i2 45	. . . 4 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
17		ancom 74	. . . . 5 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
18	17	lor 70	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = (b ∪ (b⊥ ∩ a⊥ ))
19	16, 18	ax-r2 36	. . 3 (a →2 b) = (b ∪ (b⊥ ∩ a⊥ ))
20	19	ran 78	. 2 ((a →2 b) ∩ (b ∪ c)⊥ ) = ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ )
21		df-i2 45	. 2 (a →2 c) = (c ∪ (a⊥ ∩ c⊥ ))
22	15, 20, 21	le3tr1 140	1 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ (a →2 c)

Syntax hints:   ≤ wle 2  ^⊥ 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:  3vth2  805  3vth3  806