Theorem 3vth7 810

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

Assertion

Ref	Expression
3vth7	((a →2 b)⊥ →2 (b ∪ c)) = (a →2 (b ∪ c))

Proof of Theorem 3vth7

Step	Hyp	Ref	Expression
1		df-i2 45	. . . . 5 (a →2 b) = (b ∪ (a⊥ ∩ b⊥ ))
2		df-i2 45	. . . . 5 (c →2 b) = (b ∪ (c⊥ ∩ b⊥ ))
3	1, 2	2an 79	. . . 4 ((a →2 b) ∩ (c →2 b)) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ )))
4		anass 76	. . . . . . . . . 10 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) = (a⊥ ∩ (b⊥ ∩ c⊥ ))
5	4	ax-r1 35	. . . . . . . . 9 (a⊥ ∩ (b⊥ ∩ c⊥ )) = ((a⊥ ∩ b⊥ ) ∩ c⊥ )
6		anor3 90	. . . . . . . . . 10 (b⊥ ∩ c⊥ ) = (b ∪ c)⊥
7	6	lan 77	. . . . . . . . 9 (a⊥ ∩ (b⊥ ∩ c⊥ )) = (a⊥ ∩ (b ∪ c)⊥ )
8		an32 83	. . . . . . . . 9 ((a⊥ ∩ b⊥ ) ∩ c⊥ ) = ((a⊥ ∩ c⊥ ) ∩ b⊥ )
9	5, 7, 8	3tr2 64	. . . . . . . 8 (a⊥ ∩ (b ∪ c)⊥ ) = ((a⊥ ∩ c⊥ ) ∩ b⊥ )
10		anidm 111	. . . . . . . . . 10 (b⊥ ∩ b⊥ ) = b⊥
11	10	lan 77	. . . . . . . . 9 ((a⊥ ∩ c⊥ ) ∩ (b⊥ ∩ b⊥ )) = ((a⊥ ∩ c⊥ ) ∩ b⊥ )
12	11	ax-r1 35	. . . . . . . 8 ((a⊥ ∩ c⊥ ) ∩ b⊥ ) = ((a⊥ ∩ c⊥ ) ∩ (b⊥ ∩ b⊥ ))
13		an4 86	. . . . . . . 8 ((a⊥ ∩ c⊥ ) ∩ (b⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ b⊥ ))
14	9, 12, 13	3tr 65	. . . . . . 7 (a⊥ ∩ (b ∪ c)⊥ ) = ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ b⊥ ))
15	14	lor 70	. . . . . 6 (b ∪ (a⊥ ∩ (b ∪ c)⊥ )) = (b ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ b⊥ )))
16		comanr2 465	. . . . . . . 8 b⊥ C (a⊥ ∩ b⊥ )
17	16	comcom6 459	. . . . . . 7 b C (a⊥ ∩ b⊥ )
18		comanr2 465	. . . . . . . 8 b⊥ C (c⊥ ∩ b⊥ )
19	18	comcom6 459	. . . . . . 7 b C (c⊥ ∩ b⊥ )
20	17, 19	fh3 471	. . . . . 6 (b ∪ ((a⊥ ∩ b⊥ ) ∩ (c⊥ ∩ b⊥ ))) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ )))
21	15, 20	ax-r2 36	. . . . 5 (b ∪ (a⊥ ∩ (b ∪ c)⊥ )) = ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ )))
22	21	ax-r1 35	. . . 4 ((b ∪ (a⊥ ∩ b⊥ )) ∩ (b ∪ (c⊥ ∩ b⊥ ))) = (b ∪ (a⊥ ∩ (b ∪ c)⊥ ))
23	3, 22	ax-r2 36	. . 3 ((a →2 b) ∩ (c →2 b)) = (b ∪ (a⊥ ∩ (b ∪ c)⊥ ))
24	23	lor 70	. 2 (c ∪ ((a →2 b) ∩ (c →2 b))) = (c ∪ (b ∪ (a⊥ ∩ (b ∪ c)⊥ )))
25		3vth5 808	. 2 ((a →2 b)⊥ →2 (b ∪ c)) = (c ∪ ((a →2 b) ∩ (c →2 b)))
26		df-i2 45	. . 3 (a →2 (b ∪ c)) = ((b ∪ c) ∪ (a⊥ ∩ (b ∪ c)⊥ ))
27		ax-a3 32	. . 3 ((b ∪ c) ∪ (a⊥ ∩ (b ∪ c)⊥ )) = (b ∪ (c ∪ (a⊥ ∩ (b ∪ c)⊥ )))
28		or12 80	. . 3 (b ∪ (c ∪ (a⊥ ∩ (b ∪ c)⊥ ))) = (c ∪ (b ∪ (a⊥ ∩ (b ∪ c)⊥ )))
29	26, 27, 28	3tr 65	. 2 (a →2 (b ∪ c)) = (c ∪ (b ∪ (a⊥ ∩ (b ∪ c)⊥ )))
30	24, 25, 29	3tr1 63	1 ((a →2 b)⊥ →2 (b ∪ c)) = (a →2 (b ∪ c))

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:  3vth8  811