Theorem wwfh4 211

Description: Foulis-Holland Theorem (weak).

Hypotheses

Ref	Expression
wwfh4.1	a⊥ C b
wwfh4.2	c C a

Assertion

Ref	Expression
wwfh4	((b ∪ (a ∩ c)) ≡ ((b ∪ a) ∩ (b ∪ c))) = 1

Proof of Theorem wwfh4

Step	Hyp	Ref	Expression
1		conb 114	. . 3 ((b ∪ (a ∩ c)) ≡ ((b ∪ a) ∩ (b ∪ c))) = ((b ∪ (a ∩ c))⊥ ≡ ((b ∪ a) ∩ (b ∪ c))⊥ )
2		oran 79	. . . . . 6 (b ∪ (a ∩ c)) = (b⊥ ∩ (a ∩ c)⊥ )⊥
3		df-a 39	. . . . . . . . 9 (a ∩ c) = (a⊥ ∪ c⊥ )⊥
4	3	con2 64	. . . . . . . 8 (a ∩ c)⊥ = (a⊥ ∪ c⊥ )
5	4	lan 70	. . . . . . 7 (b⊥ ∩ (a ∩ c)⊥ ) = (b⊥ ∩ (a⊥ ∪ c⊥ ))
6	5	ax-r4 36	. . . . . 6 (b⊥ ∩ (a ∩ c)⊥ )⊥ = (b⊥ ∩ (a⊥ ∪ c⊥ ))⊥
7	2, 6	ax-r2 35	. . . . 5 (b ∪ (a ∩ c)) = (b⊥ ∩ (a⊥ ∪ c⊥ ))⊥
8	7	con2 64	. . . 4 (b ∪ (a ∩ c))⊥ = (b⊥ ∩ (a⊥ ∪ c⊥ ))
9		df-a 39	. . . . . 6 ((b ∪ a) ∩ (b ∪ c)) = ((b ∪ a)⊥ ∪ (b ∪ c)⊥ )⊥
10		oran 79	. . . . . . . . 9 (b ∪ a) = (b⊥ ∩ a⊥ )⊥
11	10	con2 64	. . . . . . . 8 (b ∪ a)⊥ = (b⊥ ∩ a⊥ )
12		oran 79	. . . . . . . . 9 (b ∪ c) = (b⊥ ∩ c⊥ )⊥
13	12	con2 64	. . . . . . . 8 (b ∪ c)⊥ = (b⊥ ∩ c⊥ )
14	11, 13	2or 67	. . . . . . 7 ((b ∪ a)⊥ ∪ (b ∪ c)⊥ ) = ((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ ))
15	14	ax-r4 36	. . . . . 6 ((b ∪ a)⊥ ∪ (b ∪ c)⊥ )⊥ = ((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ ))⊥
16	9, 15	ax-r2 35	. . . . 5 ((b ∪ a) ∩ (b ∪ c)) = ((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ ))⊥
17	16	con2 64	. . . 4 ((b ∪ a) ∩ (b ∪ c))⊥ = ((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ ))
18	8, 17	2bi 91	. . 3 ((b ∪ (a ∩ c))⊥ ≡ ((b ∪ a) ∩ (b ∪ c))⊥ ) = ((b⊥ ∩ (a⊥ ∪ c⊥ )) ≡ ((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ )))
19	1, 18	ax-r2 35	. 2 ((b ∪ (a ∩ c)) ≡ ((b ∪ a) ∩ (b ∪ c))) = ((b⊥ ∩ (a⊥ ∪ c⊥ )) ≡ ((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ )))
20		wwfh4.1	. . . 4 a⊥ C b
21	20	comcom2 175	. . 3 a⊥ C b⊥
22		ax-a1 29	. . . . . 6 c = c⊥ ⊥
23	22	ax-r1 34	. . . . 5 c⊥ ⊥ = c
24		wwfh4.2	. . . . 5 c C a
25	23, 24	bctr 173	. . . 4 c⊥ ⊥ C a
26	25	comcom2 175	. . 3 c⊥ ⊥ C a⊥
27	21, 26	wwfh2 209	. 2 ((b⊥ ∩ (a⊥ ∪ c⊥ )) ≡ ((b⊥ ∩ a⊥ ) ∪ (b⊥ ∩ c⊥ ))) = 1
28	19, 27	ax-r2 35	1 ((b ∪ (a ∩ c)) ≡ ((b ∪ a) ∩ (b ∪ c))) = 1

Syntax hints:   = wb 1   C wc 3  ^⊥ wn 4   ≡ tb 5   ∪ wo 6   ∩ wa 7  1wt 9

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

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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