Theorem 3vth5 808

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

Assertion

Ref	Expression
3vth5	((a →2 b)⊥ →2 (b ∪ c)) = (c ∪ ((a →2 b) ∩ (c →2 b)))

Proof of Theorem 3vth5

Step	Hyp	Ref	Expression
1		ax-a3 32	. . 3 ((b ∪ c) ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ )) = (b ∪ (c ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ )))
2		or12 80	. . . 4 (b ∪ (c ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ ))) = (c ∪ (b ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ )))
3		comorr 184	. . . . . . 7 b C (b ∪ (a⊥ ∩ b⊥ ))
4		comorr 184	. . . . . . . 8 b C (b ∪ c)
5	4	comcom2 183	. . . . . . 7 b C (b ∪ c)⊥
6	3, 5	fh3 471	. . . . . 6 (b ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ )) = ((b ∪ (b ∪ (a⊥ ∩ b⊥ ))) ∩ (b ∪ (b ∪ c)⊥ ))
7		ax-a3 32	. . . . . . . . 9 ((b ∪ b) ∪ (a⊥ ∩ b⊥ )) = (b ∪ (b ∪ (a⊥ ∩ b⊥ )))
8	7	ax-r1 35	. . . . . . . 8 (b ∪ (b ∪ (a⊥ ∩ b⊥ ))) = ((b ∪ b) ∪ (a⊥ ∩ b⊥ ))
9		oridm 110	. . . . . . . . 9 (b ∪ b) = b
10	9	ax-r5 38	. . . . . . . 8 ((b ∪ b) ∪ (a⊥ ∩ b⊥ )) = (b ∪ (a⊥ ∩ b⊥ ))
11	8, 10	ax-r2 36	. . . . . . 7 (b ∪ (b ∪ (a⊥ ∩ b⊥ ))) = (b ∪ (a⊥ ∩ b⊥ ))
12		ancom 74	. . . . . . . . . 10 (c⊥ ∩ b⊥ ) = (b⊥ ∩ c⊥ )
13		anor3 90	. . . . . . . . . 10 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
14	12, 13	ax-r2 36	. . . . . . . . 9 (c⊥ ∩ b⊥ ) = (b ∪ c)⊥
15	14	ax-r1 35	. . . . . . . 8 (b ∪ c)⊥ = (c⊥ ∩ b⊥ )
16	15	lor 70	. . . . . . 7 (b ∪ (b ∪ c)⊥ ) = (b ∪ (c⊥ ∩ b⊥ ))
17	11, 16	2an 79	. . . . . 6 ((b ∪ (b ∪ (a⊥ ∩ b⊥ ))) ∩ (b ∪ (b ∪ c)⊥ )) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ )))
18	6, 17	ax-r2 36	. . . . 5 (b ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ )) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ )))
19	18	lor 70	. . . 4 (c ∪ (b ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ ))) = (c ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ ))))
20	2, 19	ax-r2 36	. . 3 (b ∪ (c ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ ))) = (c ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ ))))
21	1, 20	ax-r2 36	. 2 ((b ∪ c) ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ )) = (c ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ ))))
22		df-i2 45	. . 3 ((a →2 b)⊥ →2 (b ∪ c)) = ((b ∪ c) ∪ ((a →2 b)⊥ ⊥ ∩ (b ∪ c)⊥ ))
23		df-i2 45	. . . . . . . 8 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
24	23	ax-r1 35	. . . . . . 7 (b ∪ (a⊥ ∩ b⊥ )) = (a →2 b)
25		ax-a1 30	. . . . . . 7 (a →2 b) = (a →2 b)⊥ ⊥
26	24, 25	ax-r2 36	. . . . . 6 (b ∪ (a⊥ ∩ b⊥ )) = (a →2 b)⊥ ⊥
27	26	ran 78	. . . . 5 ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ ) = ((a →2 b)⊥ ⊥ ∩ (b ∪ c)⊥ )
28	27	lor 70	. . . 4 ((b ∪ c) ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ )) = ((b ∪ c) ∪ ((a →2 b)⊥ ⊥ ∩ (b ∪ c)⊥ ))
29	28	ax-r1 35	. . 3 ((b ∪ c) ∪ ((a →2 b)⊥ ⊥ ∩ (b ∪ c)⊥ )) = ((b ∪ c) ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ ))
30	22, 29	ax-r2 36	. 2 ((a →2 b)⊥ →2 (b ∪ c)) = ((b ∪ c) ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ c)⊥ ))
31		df-i2 45	. . . 4 (c →2 b) = (b ∪ (c⊥ ∩ b⊥ ))
32	23, 31	2an 79	. . 3 ((a →2 b) ∩ (c →2 b)) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ )))
33	32	lor 70	. 2 (c ∪ ((a →2 b) ∩ (c →2 b))) = (c ∪ ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ ))))
34	21, 30, 33	3tr1 63	1 ((a →2 b)⊥ →2 (b ∪ c)) = (c ∪ ((a →2 b) ∩ (c →2 b)))

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-a4 33  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  df-c1 132  df-c2 133

This theorem is referenced by:  3vth6  809  3vth7  810