Theorem u5lem6 769

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

Assertion

Ref	Expression
u5lem6	(a →5 (a →5 (a →5 b))) = (a →5 (a →5 b))

Proof of Theorem u5lem6

Step	Hyp	Ref	Expression
1		df-i5 48	. 2 (a →5 (a →5 (a →5 b))) = (((a ∩ (a →5 (a →5 b))) ∪ (a⊥ ∩ (a →5 (a →5 b)))) ∪ (a⊥ ∩ (a →5 (a →5 b))⊥ ))
2		ancom 74	. . . . 5 ((a ∪ a⊥ ) ∩ (a →5 (a →5 b))) = ((a →5 (a →5 b)) ∩ (a ∪ a⊥ ))
3		u5lemc1 684	. . . . . . 7 a C (a →5 (a →5 b))
4	3	comcom 453	. . . . . 6 (a →5 (a →5 b)) C a
5	4	comcom2 183	. . . . . 6 (a →5 (a →5 b)) C a⊥
6	4, 5	fh1r 473	. . . . 5 ((a ∪ a⊥ ) ∩ (a →5 (a →5 b))) = ((a ∩ (a →5 (a →5 b))) ∪ (a⊥ ∩ (a →5 (a →5 b))))
7		df-t 41	. . . . . . . 8 1 = (a ∪ a⊥ )
8	7	ax-r1 35	. . . . . . 7 (a ∪ a⊥ ) = 1
9	8	lan 77	. . . . . 6 ((a →5 (a →5 b)) ∩ (a ∪ a⊥ )) = ((a →5 (a →5 b)) ∩ 1)
10		an1 106	. . . . . 6 ((a →5 (a →5 b)) ∩ 1) = (a →5 (a →5 b))
11	9, 10	ax-r2 36	. . . . 5 ((a →5 (a →5 b)) ∩ (a ∪ a⊥ )) = (a →5 (a →5 b))
12	2, 6, 11	3tr2 64	. . . 4 ((a ∩ (a →5 (a →5 b))) ∪ (a⊥ ∩ (a →5 (a →5 b)))) = (a →5 (a →5 b))
13	12	ax-r5 38	. . 3 (((a ∩ (a →5 (a →5 b))) ∪ (a⊥ ∩ (a →5 (a →5 b)))) ∪ (a⊥ ∩ (a →5 (a →5 b))⊥ )) = ((a →5 (a →5 b)) ∪ (a⊥ ∩ (a →5 (a →5 b))⊥ ))
14	3	comcom3 454	. . . . 5 a⊥ C (a →5 (a →5 b))
15	3	comcom4 455	. . . . 5 a⊥ C (a →5 (a →5 b))⊥
16	14, 15	fh4 472	. . . 4 ((a →5 (a →5 b)) ∪ (a⊥ ∩ (a →5 (a →5 b))⊥ )) = (((a →5 (a →5 b)) ∪ a⊥ ) ∩ ((a →5 (a →5 b)) ∪ (a →5 (a →5 b))⊥ ))
17		df-t 41	. . . . . . 7 1 = ((a →5 (a →5 b)) ∪ (a →5 (a →5 b))⊥ )
18	17	ax-r1 35	. . . . . 6 ((a →5 (a →5 b)) ∪ (a →5 (a →5 b))⊥ ) = 1
19	18	lan 77	. . . . 5 (((a →5 (a →5 b)) ∪ a⊥ ) ∩ ((a →5 (a →5 b)) ∪ (a →5 (a →5 b))⊥ )) = (((a →5 (a →5 b)) ∪ a⊥ ) ∩ 1)
20		an1 106	. . . . . 6 (((a →5 (a →5 b)) ∪ a⊥ ) ∩ 1) = ((a →5 (a →5 b)) ∪ a⊥ )
21		u5lem5 765	. . . . . . . 8 (a →5 (a →5 b)) = (a⊥ ∪ (a ∩ b))
22	21	ax-r5 38	. . . . . . 7 ((a →5 (a →5 b)) ∪ a⊥ ) = ((a⊥ ∪ (a ∩ b)) ∪ a⊥ )
23		oridm 110	. . . . . . . . 9 (a⊥ ∪ a⊥ ) = a⊥
24	23	ax-r5 38	. . . . . . . 8 ((a⊥ ∪ a⊥ ) ∪ (a ∩ b)) = (a⊥ ∪ (a ∩ b))
25		or32 82	. . . . . . . 8 ((a⊥ ∪ (a ∩ b)) ∪ a⊥ ) = ((a⊥ ∪ a⊥ ) ∪ (a ∩ b))
26	24, 25, 21	3tr1 63	. . . . . . 7 ((a⊥ ∪ (a ∩ b)) ∪ a⊥ ) = (a →5 (a →5 b))
27	22, 26	ax-r2 36	. . . . . 6 ((a →5 (a →5 b)) ∪ a⊥ ) = (a →5 (a →5 b))
28	20, 27	ax-r2 36	. . . . 5 (((a →5 (a →5 b)) ∪ a⊥ ) ∩ 1) = (a →5 (a →5 b))
29	19, 28	ax-r2 36	. . . 4 (((a →5 (a →5 b)) ∪ a⊥ ) ∩ ((a →5 (a →5 b)) ∪ (a →5 (a →5 b))⊥ )) = (a →5 (a →5 b))
30	16, 29	ax-r2 36	. . 3 ((a →5 (a →5 b)) ∪ (a⊥ ∩ (a →5 (a →5 b))⊥ )) = (a →5 (a →5 b))
31	13, 30	ax-r2 36	. 2 (((a ∩ (a →5 (a →5 b))) ∪ (a⊥ ∩ (a →5 (a →5 b)))) ∪ (a⊥ ∩ (a →5 (a →5 b))⊥ )) = (a →5 (a →5 b))
32	1, 31	ax-r2 36	1 (a →5 (a →5 (a →5 b))) = (a →5 (a →5 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: (None)