Theorem 3vth2 787

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

Assertion

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

Proof of Theorem 3vth2

Step	Hyp	Ref	Expression
1		3vth1 786	. . 3 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ (a →2 c)
2		lear 153	. . 3 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ (b ∪ c)⊥
3	1, 2	ler2an 165	. 2 ((a →2 b) ∩ (b ∪ c)⊥ ) ≤ ((a →2 c) ∩ (b ∪ c)⊥ )
4		ax-a2 30	. . . . . 6 (b ∪ c) = (c ∪ b)
5	4	ax-r4 36	. . . . 5 (b ∪ c)⊥ = (c ∪ b)⊥
6	5	lan 70	. . . 4 ((a →2 c) ∩ (b ∪ c)⊥ ) = ((a →2 c) ∩ (c ∪ b)⊥ )
7		3vth1 786	. . . 4 ((a →2 c) ∩ (c ∪ b)⊥ ) ≤ (a →2 b)
8	6, 7	bltr 130	. . 3 ((a →2 c) ∩ (b ∪ c)⊥ ) ≤ (a →2 b)
9		lear 153	. . 3 ((a →2 c) ∩ (b ∪ c)⊥ ) ≤ (b ∪ c)⊥
10	8, 9	ler2an 165	. 2 ((a →2 c) ∩ (b ∪ c)⊥ ) ≤ ((a →2 b) ∩ (b ∪ c)⊥ )
11	3, 10	lebi 137	1 ((a →2 b) ∩ (b ∪ c)⊥ ) = ((a →2 c) ∩ (b ∪ c)⊥ )

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 14

This theorem is referenced by:  3vth4 789

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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