Theorem wfh4 426

Description: Weak structural analog of Foulis-Holland Theorem. (Contributed by NM, 13-Oct-1997.)

Hypotheses

Ref	Expression
wfh.1	C (a, b) = 1
wfh.2	C (a, c) = 1

Assertion

Ref	Expression
wfh4	((b ∪ (a ∩ c)) ≡ ((b ∪ a) ∩ (b ∪ c))) = 1

Proof of Theorem wfh4

Step	Hyp	Ref	Expression
1		wfh.1	. . . . 5 C (a, b) = 1
2	1	wcomcom4 417	. . . 4 C (a⊥ , b⊥ ) = 1
3		wfh.2	. . . . 5 C (a, c) = 1
4	3	wcomcom4 417	. . . 4 C (a⊥ , c⊥ ) = 1
5	2, 4	wfh2 424	. . 3 ((b⊥ ∩ (a⊥ ∪ c⊥ )) ≡ ((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ ))) = 1
6		anor2 89	. . . . 5 (b⊥ ∩ (a⊥ ∪ c⊥ )) = (b ∪ (a⊥ ∪ c⊥ )⊥ )⊥
7	6	bi1 118	. . . 4 ((b⊥ ∩ (a⊥ ∪ c⊥ )) ≡ (b ∪ (a⊥ ∪ c⊥ )⊥ )⊥ ) = 1
8		df-a 40	. . . . . . . 8 (a ∩ c) = (a⊥ ∪ c⊥ )⊥
9	8	bi1 118	. . . . . . 7 ((a ∩ c) ≡ (a⊥ ∪ c⊥ )⊥ ) = 1
10	9	wr1 197	. . . . . 6 ((a⊥ ∪ c⊥ )⊥ ≡ (a ∩ c)) = 1
11	10	wlor 368	. . . . 5 ((b ∪ (a⊥ ∪ c⊥ )⊥ ) ≡ (b ∪ (a ∩ c))) = 1
12	11	wr4 199	. . . 4 ((b ∪ (a⊥ ∪ c⊥ )⊥ )⊥ ≡ (b ∪ (a ∩ c))⊥ ) = 1
13	7, 12	wr2 371	. . 3 ((b⊥ ∩ (a⊥ ∪ c⊥ )) ≡ (b ∪ (a ∩ c))⊥ ) = 1
14		oran 87	. . . . 5 ((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ )) = ((b⊥ ∩ a⊥ )⊥ ∩ (b⊥ ∩ c⊥ )⊥ )⊥
15	14	bi1 118	. . . 4 (((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ )) ≡ ((b⊥ ∩ a⊥ )⊥ ∩ (b⊥ ∩ c⊥ )⊥ )⊥ ) = 1
16		oran 87	. . . . . . . 8 (b ∪ a) = (b⊥ ∩ a⊥ )⊥
17	16	bi1 118	. . . . . . 7 ((b ∪ a) ≡ (b⊥ ∩ a⊥ )⊥ ) = 1
18		oran 87	. . . . . . . 8 (b ∪ c) = (b⊥ ∩ c⊥ )⊥
19	18	bi1 118	. . . . . . 7 ((b ∪ c) ≡ (b⊥ ∩ c⊥ )⊥ ) = 1
20	17, 19	w2an 373	. . . . . 6 (((b ∪ a) ∩ (b ∪ c)) ≡ ((b⊥ ∩ a⊥ )⊥ ∩ (b⊥ ∩ c⊥ )⊥ )) = 1
21	20	wr1 197	. . . . 5 (((b⊥ ∩ a⊥ )⊥ ∩ (b⊥ ∩ c⊥ )⊥ ) ≡ ((b ∪ a) ∩ (b ∪ c))) = 1
22	21	wr4 199	. . . 4 (((b⊥ ∩ a⊥ )⊥ ∩ (b⊥ ∩ c⊥ )⊥ )⊥ ≡ ((b ∪ a) ∩ (b ∪ c))⊥ ) = 1
23	15, 22	wr2 371	. . 3 (((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ )) ≡ ((b ∪ a) ∩ (b ∪ c))⊥ ) = 1
24	5, 13, 23	w3tr2 375	. 2 ((b ∪ (a ∩ c))⊥ ≡ ((b ∪ a) ∩ (b ∪ c))⊥ ) = 1
25	24	wcon1 207	1 ((b ∪ (a ∩ c)) ≡ ((b ∪ a) ∩ (b ∪ c))) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 8   C wcmtr 29

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-wom 361

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i1 44  df-i2 45  df-le 129  df-le1 130  df-le2 131  df-cmtr 134

This theorem is referenced by:  ska2  432  ska4  433  woml6  436