Theorem 3vth1 786

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

Assertion

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

Proof of Theorem 3vth1

Step	Hyp	Ref	Expression
1		anor3 82	. . . . . . 7 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
2	1	lan 70	. . . . . 6 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∩ c⊥ )) = ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ )
3	2	ax-r1 34	. . . . 5 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ ) = ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∩ c⊥ ))
4		anass 69	. . . . . 6 (((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ∩ c⊥ ) = ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∩ c⊥ ))
5	4	ax-r1 34	. . . . 5 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b⊥ ∩ c⊥ )) = (((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ∩ c⊥ )
6	3, 5	ax-r2 35	. . . 4 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ ) = (((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ∩ c⊥ )
7		ancom 68	. . . . . . 7 ((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) = (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ )))
8		omlan 430	. . . . . . 7 (b⊥ ∩ (b ∪ (b⊥ ∩ a⊥ ))) = (b⊥ ∩ a⊥ )
9	7, 8	ax-r2 35	. . . . . 6 ((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) = (b⊥ ∩ a⊥ )
10		lear 153	. . . . . 6 (b⊥ ∩ a⊥ ) ≤ a⊥
11	9, 10	bltr 130	. . . . 5 ((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ≤ a⊥
12	11	leran 145	. . . 4 (((b ∪ (b⊥ ∩ a⊥ )) ∩ b⊥ ) ∩ c⊥ ) ≤ (a⊥ ∩ c⊥ )
13	6, 12	bltr 130	. . 3 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ ) ≤ (a⊥ ∩ c⊥ )
14		leor 151	. . 3 (a⊥ ∩ c⊥ ) ≤ (c ∪ (a⊥ ∩ c⊥ ))
15	13, 14	letr 129	. 2 ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ ) ≤ (c ∪ (a⊥ ∩ c⊥ ))
16		df-i2 44	. . . 4 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
17		ancom 68	. . . . 5 (a⊥ ∩ b⊥ ) = (b⊥ ∩ a⊥ )
18	17	lor 66	. . . 4 (b ∪ (a⊥ ∩ b⊥ )) = (b ∪ (b⊥ ∩ a⊥ ))
19	16, 18	ax-r2 35	. . 3 (a →2 b) = (b ∪ (b⊥ ∩ a⊥ ))
20	19	ran 71	. 2 ((a →2 b) ∩ (b ∪ c)⊥ ) = ((b ∪ (b⊥ ∩ a⊥ )) ∩ (b ∪ c)⊥ )
21		df-i2 44	. 2 (a →2 c) = (c ∪ (a⊥ ∩ c⊥ ))
22	15, 20, 21	le3tr1 132	1 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ (a →2 c)

Syntax hints:   ≤ wle 2  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 14

This theorem is referenced by:  3vth2 787  3vth3 788

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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