Theorem ud4lem0c 280

Description: Lemma for unified disjunction. (Contributed by NM, 23-Nov-1997.)

Assertion

Ref	Expression
ud4lem0c	(a →4 b)⊥ = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a ∩ b⊥ ) ∪ b))

Proof of Theorem ud4lem0c

Step	Hyp	Ref	Expression
1		df-i4 47	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2		oran 87	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (((a ∩ b) ∪ (a⊥ ∩ b))⊥ ∩ ((a⊥ ∪ b) ∩ b⊥ )⊥ )⊥
3		oran 87	. . . . . . . 8 ((a ∩ b) ∪ (a⊥ ∩ b)) = ((a ∩ b)⊥ ∩ (a⊥ ∩ b)⊥ )⊥
4		df-a 40	. . . . . . . . . . 11 (a ∩ b) = (a⊥ ∪ b⊥ )⊥
5	4	con2 67	. . . . . . . . . 10 (a ∩ b)⊥ = (a⊥ ∪ b⊥ )
6		anor2 89	. . . . . . . . . . 11 (a⊥ ∩ b) = (a ∪ b⊥ )⊥
7	6	con2 67	. . . . . . . . . 10 (a⊥ ∩ b)⊥ = (a ∪ b⊥ )
8	5, 7	2an 79	. . . . . . . . 9 ((a ∩ b)⊥ ∩ (a⊥ ∩ b)⊥ ) = ((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ ))
9	8	ax-r4 37	. . . . . . . 8 ((a ∩ b)⊥ ∩ (a⊥ ∩ b)⊥ )⊥ = ((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ ))⊥
10	3, 9	ax-r2 36	. . . . . . 7 ((a ∩ b) ∪ (a⊥ ∩ b)) = ((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ ))⊥
11	10	con2 67	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b))⊥ = ((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ ))
12		anor1 88	. . . . . . . 8 ((a⊥ ∪ b) ∩ b⊥ ) = ((a⊥ ∪ b)⊥ ∪ b)⊥
13		anor1 88	. . . . . . . . . . 11 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
14	13	ax-r1 35	. . . . . . . . . 10 (a⊥ ∪ b)⊥ = (a ∩ b⊥ )
15	14	ax-r5 38	. . . . . . . . 9 ((a⊥ ∪ b)⊥ ∪ b) = ((a ∩ b⊥ ) ∪ b)
16	15	ax-r4 37	. . . . . . . 8 ((a⊥ ∪ b)⊥ ∪ b)⊥ = ((a ∩ b⊥ ) ∪ b)⊥
17	12, 16	ax-r2 36	. . . . . . 7 ((a⊥ ∪ b) ∩ b⊥ ) = ((a ∩ b⊥ ) ∪ b)⊥
18	17	con2 67	. . . . . 6 ((a⊥ ∪ b) ∩ b⊥ )⊥ = ((a ∩ b⊥ ) ∪ b)
19	11, 18	2an 79	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b))⊥ ∩ ((a⊥ ∪ b) ∩ b⊥ )⊥ ) = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a ∩ b⊥ ) ∪ b))
20	19	ax-r4 37	. . . 4 (((a ∩ b) ∪ (a⊥ ∩ b))⊥ ∩ ((a⊥ ∪ b) ∩ b⊥ )⊥ )⊥ = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a ∩ b⊥ ) ∪ b))⊥
21	2, 20	ax-r2 36	. . 3 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a ∩ b⊥ ) ∪ b))⊥
22	1, 21	ax-r2 36	. 2 (a →4 b) = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a ∩ b⊥ ) ∪ b))⊥
23	22	con2 67	1 (a →4 b)⊥ = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a ∩ b⊥ ) ∪ b))

This proof depends on syntax axioms:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₄ wi4 15

This proof depends on axioms:  ax-a1 30  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This proof depends on definitions:  df-a 40  df-i4 47

This theorem is used by:  ud4lem1b  578  ud4lem1c  579  ud4lem1d  580  ud4lem3a  583  ud4lem3b  584  u4lem5  764