Theorem ud5lem1c 588

Description: Lemma for unified disjunction. (Contributed by NM, 26-Nov-1997.)

Assertion

Ref	Expression
ud5lem1c	((a →5 b)⊥ ∩ (b →5 a)⊥ ) = (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))

Proof of Theorem ud5lem1c

Step	Hyp	Ref	Expression
1		ud5lem0c 281	. . 3 (a →5 b)⊥ = (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b))
2		ud5lem0c 281	. . . 4 (b →5 a)⊥ = (((b⊥ ∪ a⊥ ) ∩ (b ∪ a⊥ )) ∩ (b ∪ a))
3		ax-a2 31	. . . . . 6 (b⊥ ∪ a⊥ ) = (a⊥ ∪ b⊥ )
4		ax-a2 31	. . . . . 6 (b ∪ a⊥ ) = (a⊥ ∪ b)
5	3, 4	2an 79	. . . . 5 ((b⊥ ∪ a⊥ ) ∩ (b ∪ a⊥ )) = ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))
6		ax-a2 31	. . . . 5 (b ∪ a) = (a ∪ b)
7	5, 6	2an 79	. . . 4 (((b⊥ ∪ a⊥ ) ∩ (b ∪ a⊥ )) ∩ (b ∪ a)) = (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)) ∩ (a ∪ b))
8	2, 7	ax-r2 36	. . 3 (b →5 a)⊥ = (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)) ∩ (a ∪ b))
9	1, 8	2an 79	. 2 ((a →5 b)⊥ ∩ (b →5 a)⊥ ) = ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)) ∩ (a ∪ b)))
10		an4 86	. . 3 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)) ∩ (a ∪ b))) = ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))) ∩ ((a ∪ b) ∩ (a ∪ b)))
11		ancom 74	. . . 4 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))) ∩ ((a ∪ b) ∩ (a ∪ b))) = (((a ∪ b) ∩ (a ∪ b)) ∩ (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))))
12		anidm 111	. . . . . 6 ((a ∪ b) ∩ (a ∪ b)) = (a ∪ b)
13		an4 86	. . . . . . 7 (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))) = (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b⊥ )) ∩ ((a ∪ b⊥ ) ∩ (a⊥ ∪ b)))
14		anidm 111	. . . . . . . . . 10 ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b⊥ )) = (a⊥ ∪ b⊥ )
15	14	ran 78	. . . . . . . . 9 (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b⊥ )) ∩ ((a ∪ b⊥ ) ∩ (a⊥ ∪ b))) = ((a⊥ ∪ b⊥ ) ∩ ((a ∪ b⊥ ) ∩ (a⊥ ∪ b)))
16		ancom 74	. . . . . . . . 9 ((a⊥ ∪ b⊥ ) ∩ ((a ∪ b⊥ ) ∩ (a⊥ ∪ b))) = (((a ∪ b⊥ ) ∩ (a⊥ ∪ b)) ∩ (a⊥ ∪ b⊥ ))
17	15, 16	ax-r2 36	. . . . . . . 8 (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b⊥ )) ∩ ((a ∪ b⊥ ) ∩ (a⊥ ∪ b))) = (((a ∪ b⊥ ) ∩ (a⊥ ∪ b)) ∩ (a⊥ ∪ b⊥ ))
18		anass 76	. . . . . . . 8 (((a ∪ b⊥ ) ∩ (a⊥ ∪ b)) ∩ (a⊥ ∪ b⊥ )) = ((a ∪ b⊥ ) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))
19	17, 18	ax-r2 36	. . . . . . 7 (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b⊥ )) ∩ ((a ∪ b⊥ ) ∩ (a⊥ ∪ b))) = ((a ∪ b⊥ ) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))
20	13, 19	ax-r2 36	. . . . . 6 (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))) = ((a ∪ b⊥ ) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))
21	12, 20	2an 79	. . . . 5 (((a ∪ b) ∩ (a ∪ b)) ∩ (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)))) = ((a ∪ b) ∩ ((a ∪ b⊥ ) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))))
22		anass 76	. . . . . 6 (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))) = ((a ∪ b) ∩ ((a ∪ b⊥ ) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ ))))
23	22	ax-r1 35	. . . . 5 ((a ∪ b) ∩ ((a ∪ b⊥ ) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))) = (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))
24	21, 23	ax-r2 36	. . . 4 (((a ∪ b) ∩ (a ∪ b)) ∩ (((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)))) = (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))
25	11, 24	ax-r2 36	. . 3 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b))) ∩ ((a ∪ b) ∩ (a ∪ b))) = (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))
26	10, 25	ax-r2 36	. 2 ((((a⊥ ∪ b⊥ ) ∩ (a ∪ b⊥ )) ∩ (a ∪ b)) ∩ (((a⊥ ∪ b⊥ ) ∩ (a⊥ ∪ b)) ∩ (a ∪ b))) = (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))
27	9, 26	ax-r2 36	1 ((a →5 b)⊥ ∩ (b →5 a)⊥ ) = (((a ∪ b) ∩ (a ∪ b⊥ )) ∩ ((a⊥ ∪ b) ∩ (a⊥ ∪ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₅ wi5 16

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i5 48

This theorem is referenced by:  ud5lem1  589