Theorem wwfh3 218

Description: Foulis-Holland Theorem (weak). (Contributed by NM, 3-Sep-1997.)

Hypotheses

Ref	Expression
wwfh3.1	b⊥ C a
wwfh3.2	c⊥ C a

Assertion

Ref	Expression
wwfh3	((a ∪ (b ∩ c)) ≡ ((a ∪ b) ∩ (a ∪ c))) = 1

Proof of Theorem wwfh3

Step	Hyp	Ref	Expression
1		conb 122	. . 3 ((a ∪ (b ∩ c)) ≡ ((a ∪ b) ∩ (a ∪ c))) = ((a ∪ (b ∩ c))⊥ ≡ ((a ∪ b) ∩ (a ∪ c))⊥ )
2		oran 87	. . . . . 6 (a ∪ (b ∩ c)) = (a⊥ ∩ (b ∩ c)⊥ )⊥
3		df-a 40	. . . . . . . . 9 (b ∩ c) = (b⊥ ∪ c⊥ )⊥
4	3	con2 67	. . . . . . . 8 (b ∩ c)⊥ = (b⊥ ∪ c⊥ )
5	4	lan 77	. . . . . . 7 (a⊥ ∩ (b ∩ c)⊥ ) = (a⊥ ∩ (b⊥ ∪ c⊥ ))
6	5	ax-r4 37	. . . . . 6 (a⊥ ∩ (b ∩ c)⊥ )⊥ = (a⊥ ∩ (b⊥ ∪ c⊥ ))⊥
7	2, 6	ax-r2 36	. . . . 5 (a ∪ (b ∩ c)) = (a⊥ ∩ (b⊥ ∪ c⊥ ))⊥
8	7	con2 67	. . . 4 (a ∪ (b ∩ c))⊥ = (a⊥ ∩ (b⊥ ∪ c⊥ ))
9		df-a 40	. . . . . 6 ((a ∪ b) ∩ (a ∪ c)) = ((a ∪ b)⊥ ∪ (a ∪ c)⊥ )⊥
10		oran 87	. . . . . . . . 9 (a ∪ b) = (a⊥ ∩ b⊥ )⊥
11	10	con2 67	. . . . . . . 8 (a ∪ b)⊥ = (a⊥ ∩ b⊥ )
12		oran 87	. . . . . . . . 9 (a ∪ c) = (a⊥ ∩ c⊥ )⊥
13	12	con2 67	. . . . . . . 8 (a ∪ c)⊥ = (a⊥ ∩ c⊥ )
14	11, 13	2or 72	. . . . . . 7 ((a ∪ b)⊥ ∪ (a ∪ c)⊥ ) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ ))
15	14	ax-r4 37	. . . . . 6 ((a ∪ b)⊥ ∪ (a ∪ c)⊥ )⊥ = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ ))⊥
16	9, 15	ax-r2 36	. . . . 5 ((a ∪ b) ∩ (a ∪ c)) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ ))⊥
17	16	con2 67	. . . 4 ((a ∪ b) ∩ (a ∪ c))⊥ = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ ))
18	8, 17	2bi 99	. . 3 ((a ∪ (b ∩ c))⊥ ≡ ((a ∪ b) ∩ (a ∪ c))⊥ ) = ((a⊥ ∩ (b⊥ ∪ c⊥ )) ≡ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ )))
19	1, 18	ax-r2 36	. 2 ((a ∪ (b ∩ c)) ≡ ((a ∪ b) ∩ (a ∪ c))) = ((a⊥ ∩ (b⊥ ∪ c⊥ )) ≡ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ )))
20		wwfh3.1	. . . 4 b⊥ C a
21	20	comcom2 183	. . 3 b⊥ C a⊥
22		wwfh3.2	. . . 4 c⊥ C a
23	22	comcom2 183	. . 3 c⊥ C a⊥
24	21, 23	wwfh1 216	. 2 ((a⊥ ∩ (b⊥ ∪ c⊥ )) ≡ ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ c⊥ ))) = 1
25	19, 24	ax-r2 36	1 ((a ∪ (b ∩ c)) ≡ ((a ∪ b) ∩ (a ∪ c))) = 1

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8

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

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)