Theorem u4lem1n 742

Description: Lemma for unified implication study. (Contributed by NM, 16-Dec-1997.)

Assertion

Ref	Expression
u4lem1n	((a →4 b) →4 a)⊥ = ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Proof of Theorem u4lem1n

Step	Hyp	Ref	Expression
1		oran1 91	. . . . 5 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) = (((a ∩ b) ∪ (a ∩ b⊥ ))⊥ ∩ a)⊥
2		df-a 40	. . . . . . . . . . 11 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
3		anor1 88	. . . . . . . . . . 11 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
4	2, 3	2or 72	. . . . . . . . . 10 ((a ∩ b) ∪ (a ∩ b⊥ )) = ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ b)⊥ )
5	4	ax-r4 37	. . . . . . . . 9 ((a ∩ b) ∪ (a ∩ b⊥ ))⊥ = ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ b)⊥ )⊥
6		df-a 40	. . . . . . . . . 10 ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ b)⊥ )⊥
7	6	ax-r1 35	. . . . . . . . 9 ((a⊥ ∪ b⊥ )⊥ ∪ (a⊥ ∪ b)⊥ )⊥ = ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))
8	5, 7	ax-r2 36	. . . . . . . 8 ((a ∩ b) ∪ (a ∩ b⊥ ))⊥ = ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))
9		ancom 74	. . . . . . . 8 ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)) = ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))
10	8, 9	ax-r2 36	. . . . . . 7 ((a ∩ b) ∪ (a ∩ b⊥ ))⊥ = ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))
11	10	ran 78	. . . . . 6 (((a ∩ b) ∪ (a ∩ b⊥ ))⊥ ∩ a) = (((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a)
12	11	ax-r4 37	. . . . 5 (((a ∩ b) ∪ (a ∩ b⊥ ))⊥ ∩ a)⊥ = (((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a)⊥
13	1, 12	ax-r2 36	. . . 4 (((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) = (((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a)⊥
14		ancom 74	. . . . 5 ((a ∪ b) ∩ (a ∪ b⊥ )) = ((a ∪ b⊥ ) ∩ (a ∪ b))
15		df-a 40	. . . . . 6 ((a ∪ b⊥ ) ∩ (a ∪ b)) = ((a ∪ b⊥ )⊥ ∪ (a ∪ b)⊥ )⊥
16		anor2 89	. . . . . . . . 9 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
17		anor3 90	. . . . . . . . 9 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
18	16, 17	2or 72	. . . . . . . 8 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) = ((a ∪ b⊥ )⊥ ∪ (a ∪ b)⊥ )
19	18	ax-r4 37	. . . . . . 7 ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ = ((a ∪ b⊥ )⊥ ∪ (a ∪ b)⊥ )⊥
20	19	ax-r1 35	. . . . . 6 ((a ∪ b⊥ )⊥ ∪ (a ∪ b)⊥ )⊥ = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥
21	15, 20	ax-r2 36	. . . . 5 ((a ∪ b⊥ ) ∩ (a ∪ b)) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥
22	14, 21	ax-r2 36	. . . 4 ((a ∪ b) ∩ (a ∪ b⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥
23	13, 22	2an 79	. . 3 ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ ))) = ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a)⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )
24	23	ax-r4 37	. 2 ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))⊥ = ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a)⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )⊥
25		u4lem1 737	. . 3 ((a →4 b) →4 a) = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))
26	25	ax-r4 37	. 2 ((a →4 b) →4 a)⊥ = ((((a ∩ b) ∪ (a ∩ b⊥ )) ∪ a⊥ ) ∩ ((a ∪ b) ∩ (a ∪ b⊥ )))⊥
27		oran 87	. 2 ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))) = ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a)⊥ ∩ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))⊥ )⊥
28	24, 26, 27	3tr1 63	1 ((a →4 b) →4 a)⊥ = ((((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )) ∩ a) ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 15

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-i4 47  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u4lem2  747