Theorem u5lem5 765

Description: Lemma for unified implication study. (Contributed by NM, 20-Dec-1997.)

Assertion

Ref	Expression
u5lem5	(a →5 (a →5 b)) = (a⊥ ∪ (a ∩ b))

Proof of Theorem u5lem5

Step	Hyp	Ref	Expression
1		df-i5 48	. 2 (a →5 (a →5 b)) = (((a ∩ (a →5 b)) ∪ (a⊥ ∩ (a →5 b))) ∪ (a⊥ ∩ (a →5 b)⊥ ))
2		u5lemc1 684	. . . . . . . 8 a C (a →5 b)
3	2	comcom 453	. . . . . . 7 (a →5 b) C a
4	3	comcom2 183	. . . . . . 7 (a →5 b) C a⊥
5	3, 4	fh1r 473	. . . . . 6 ((a ∪ a⊥ ) ∩ (a →5 b)) = ((a ∩ (a →5 b)) ∪ (a⊥ ∩ (a →5 b)))
6	5	ax-r1 35	. . . . 5 ((a ∩ (a →5 b)) ∪ (a⊥ ∩ (a →5 b))) = ((a ∪ a⊥ ) ∩ (a →5 b))
7		ancom 74	. . . . . 6 ((a ∪ a⊥ ) ∩ (a →5 b)) = ((a →5 b) ∩ (a ∪ a⊥ ))
8		df-t 41	. . . . . . . . 9 1 = (a ∪ a⊥ )
9	8	ax-r1 35	. . . . . . . 8 (a ∪ a⊥ ) = 1
10	9	lan 77	. . . . . . 7 ((a →5 b) ∩ (a ∪ a⊥ )) = ((a →5 b) ∩ 1)
11		an1 106	. . . . . . 7 ((a →5 b) ∩ 1) = (a →5 b)
12	10, 11	ax-r2 36	. . . . . 6 ((a →5 b) ∩ (a ∪ a⊥ )) = (a →5 b)
13	7, 12	ax-r2 36	. . . . 5 ((a ∪ a⊥ ) ∩ (a →5 b)) = (a →5 b)
14	6, 13	ax-r2 36	. . . 4 ((a ∩ (a →5 b)) ∪ (a⊥ ∩ (a →5 b))) = (a →5 b)
15	14	ax-r5 38	. . 3 (((a ∩ (a →5 b)) ∪ (a⊥ ∩ (a →5 b))) ∪ (a⊥ ∩ (a →5 b)⊥ )) = ((a →5 b) ∪ (a⊥ ∩ (a →5 b)⊥ ))
16	2	comcom3 454	. . . . 5 a⊥ C (a →5 b)
17	2	comcom4 455	. . . . 5 a⊥ C (a →5 b)⊥
18	16, 17	fh4 472	. . . 4 ((a →5 b) ∪ (a⊥ ∩ (a →5 b)⊥ )) = (((a →5 b) ∪ a⊥ ) ∩ ((a →5 b) ∪ (a →5 b)⊥ ))
19		df-t 41	. . . . . . 7 1 = ((a →5 b) ∪ (a →5 b)⊥ )
20	19	ax-r1 35	. . . . . 6 ((a →5 b) ∪ (a →5 b)⊥ ) = 1
21	20	lan 77	. . . . 5 (((a →5 b) ∪ a⊥ ) ∩ ((a →5 b) ∪ (a →5 b)⊥ )) = (((a →5 b) ∪ a⊥ ) ∩ 1)
22		an1 106	. . . . . 6 (((a →5 b) ∪ a⊥ ) ∩ 1) = ((a →5 b) ∪ a⊥ )
23		u5lemona 629	. . . . . 6 ((a →5 b) ∪ a⊥ ) = (a⊥ ∪ (a ∩ b))
24	22, 23	ax-r2 36	. . . . 5 (((a →5 b) ∪ a⊥ ) ∩ 1) = (a⊥ ∪ (a ∩ b))
25	21, 24	ax-r2 36	. . . 4 (((a →5 b) ∪ a⊥ ) ∩ ((a →5 b) ∪ (a →5 b)⊥ )) = (a⊥ ∪ (a ∩ b))
26	18, 25	ax-r2 36	. . 3 ((a →5 b) ∪ (a⊥ ∩ (a →5 b)⊥ )) = (a⊥ ∪ (a ∩ b))
27	15, 26	ax-r2 36	. 2 (((a ∩ (a →5 b)) ∪ (a⊥ ∩ (a →5 b))) ∪ (a⊥ ∩ (a →5 b)⊥ )) = (a⊥ ∪ (a ∩ b))
28	1, 27	ax-r2 36	1 (a →5 (a →5 b)) = (a⊥ ∪ (a ∩ b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₅ wi5 16

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-i5 48  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u5lem6  769