Theorem 3vth9 794

Description: A 3-variable theorem.

Assertion

Ref	Expression
3vth9	((a ∪ b) →1 (c →2 b)) = ((b ∪ c) →2 (a →2 b))

Proof of Theorem 3vth9

Step	Hyp	Ref	Expression
1		anor3 82	. . . 4 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
2	1	ax-r1 34	. . 3 (a ∪ b)⊥ = (a⊥ ∩ b⊥ )
3		df-i2 44	. . . 4 (c →2 b) = (b ∪ (c⊥ ∩ b⊥ ))
4	3	lan 70	. . 3 ((a ∪ b) ∩ (c →2 b)) = ((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ )))
5	2, 4	2or 67	. 2 ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ (c →2 b))) = ((a⊥ ∩ b⊥ ) ∪ ((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ ))))
6		df-i1 43	. 2 ((a ∪ b) →1 (c →2 b)) = ((a ∪ b)⊥ ∪ ((a ∪ b) ∩ (c →2 b)))
7		df-i2 44	. . . 4 ((b ∪ c) →2 (a →2 b)) = ((a →2 b) ∪ ((b ∪ c)⊥ ∩ (a →2 b)⊥ ))
8		df-i2 44	. . . . 5 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
9		anor3 82	. . . . . . . 8 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
10	9	ax-r1 34	. . . . . . 7 (b ∪ c)⊥ = (b⊥ ∩ c⊥ )
11		ud2lem0c 270	. . . . . . 7 (a →2 b)⊥ = (b⊥ ∩ (a ∪ b))
12	10, 11	2an 72	. . . . . 6 ((b ∪ c)⊥ ∩ (a →2 b)⊥ ) = ((b⊥ ∩ c⊥ ) ∩ (b⊥ ∩ (a ∪ b)))
13		anandi 106	. . . . . . . 8 (b⊥ ∩ (c⊥ ∩ (a ∪ b))) = ((b⊥ ∩ c⊥ ) ∩ (b⊥ ∩ (a ∪ b)))
14	13	ax-r1 34	. . . . . . 7 ((b⊥ ∩ c⊥ ) ∩ (b⊥ ∩ (a ∪ b))) = (b⊥ ∩ (c⊥ ∩ (a ∪ b)))
15		anass 69	. . . . . . . 8 ((b⊥ ∩ c⊥ ) ∩ (a ∪ b)) = (b⊥ ∩ (c⊥ ∩ (a ∪ b)))
16	15	ax-r1 34	. . . . . . 7 (b⊥ ∩ (c⊥ ∩ (a ∪ b))) = ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))
17	14, 16	ax-r2 35	. . . . . 6 ((b⊥ ∩ c⊥ ) ∩ (b⊥ ∩ (a ∪ b))) = ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))
18	12, 17	ax-r2 35	. . . . 5 ((b ∪ c)⊥ ∩ (a →2 b)⊥ ) = ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))
19	8, 18	2or 67	. . . 4 ((a →2 b) ∪ ((b ∪ c)⊥ ∩ (a →2 b)⊥ )) = ((b ∪ (a⊥ ∩ b⊥ )) ∪ ((b⊥ ∩ c⊥ ) ∩ (a ∪ b)))
20	7, 19	ax-r2 35	. . 3 ((b ∪ c) →2 (a →2 b)) = ((b ∪ (a⊥ ∩ b⊥ )) ∪ ((b⊥ ∩ c⊥ ) ∩ (a ∪ b)))
21		or32 75	. . . 4 ((b ∪ (a⊥ ∩ b⊥ )) ∪ ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))) = ((b ∪ ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))) ∪ (a⊥ ∩ b⊥ ))
22		comanr1 446	. . . . . . . . 9 b⊥ C (b⊥ ∩ c⊥ )
23	22	comcom6 441	. . . . . . . 8 b C (b⊥ ∩ c⊥ )
24		comorr2 445	. . . . . . . 8 b C (a ∪ b)
25	23, 24	fh3 453	. . . . . . 7 (b ∪ ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))) = ((b ∪ (b⊥ ∩ c⊥ )) ∩ (b ∪ (a ∪ b)))
26		ancom 68	. . . . . . . . . 10 (b⊥ ∩ c⊥ ) = (c⊥ ∩ b⊥ )
27	26	lor 66	. . . . . . . . 9 (b ∪ (b⊥ ∩ c⊥ )) = (b ∪ (c⊥ ∩ b⊥ ))
28		or12 73	. . . . . . . . . 10 (b ∪ (a ∪ b)) = (a ∪ (b ∪ b))
29		oridm 102	. . . . . . . . . . 11 (b ∪ b) = b
30	29	lor 66	. . . . . . . . . 10 (a ∪ (b ∪ b)) = (a ∪ b)
31	28, 30	ax-r2 35	. . . . . . . . 9 (b ∪ (a ∪ b)) = (a ∪ b)
32	27, 31	2an 72	. . . . . . . 8 ((b ∪ (b⊥ ∩ c⊥ )) ∩ (b ∪ (a ∪ b))) = ((b ∪ (c⊥ ∩ b⊥ )) ∩ (a ∪ b))
33		ancom 68	. . . . . . . 8 ((b ∪ (c⊥ ∩ b⊥ )) ∩ (a ∪ b)) = ((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ )))
34	32, 33	ax-r2 35	. . . . . . 7 ((b ∪ (b⊥ ∩ c⊥ )) ∩ (b ∪ (a ∪ b))) = ((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ )))
35	25, 34	ax-r2 35	. . . . . 6 (b ∪ ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))) = ((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ )))
36	35	ax-r5 37	. . . . 5 ((b ∪ ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))) ∪ (a⊥ ∩ b⊥ )) = (((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ ))) ∪ (a⊥ ∩ b⊥ ))
37		ax-a2 30	. . . . 5 (((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ ))) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ ((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ ))))
38	36, 37	ax-r2 35	. . . 4 ((b ∪ ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ ((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ ))))
39	21, 38	ax-r2 35	. . 3 ((b ∪ (a⊥ ∩ b⊥ )) ∪ ((b⊥ ∩ c⊥ ) ∩ (a ∪ b))) = ((a⊥ ∩ b⊥ ) ∪ ((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ ))))
40	20, 39	ax-r2 35	. 2 ((b ∪ c) →2 (a →2 b)) = ((a⊥ ∩ b⊥ ) ∪ ((a ∪ b) ∩ (b ∪ (c⊥ ∩ b⊥ ))))
41	5, 6, 40	3tr1 60	1 ((a ∪ b) →1 (c →2 b)) = ((b ∪ c) →2 (a →2 b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₁ wi1 13   →₂ wi2 14

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  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-i1 43  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

metamath.org