Theorem u4lemob 633

Description: Lemma for non-tollens implication study. (Contributed by NM, 15-Dec-1997.)

Assertion

Ref	Expression
u4lemob	((a →4 b) ∪ b) = (a⊥ ∪ b)

Proof of Theorem u4lemob

Step	Hyp	Ref	Expression
1		df-i4 47	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2	1	ax-r5 38	. 2 ((a →4 b) ∪ b) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ b)
3		or32 82	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ b) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
4		lear 161	. . . . . . 7 (a ∩ b) ≤ b
5		lear 161	. . . . . . 7 (a⊥ ∩ b) ≤ b
6	4, 5	lel2or 170	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b)) ≤ b
7	6	df-le2 131	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) = b
8	7	ax-r5 38	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (b ∪ ((a⊥ ∪ b) ∩ b⊥ ))
9		comorr2 463	. . . . . 6 b C (a⊥ ∪ b)
10		comid 187	. . . . . . 7 b C b
11	10	comcom2 183	. . . . . 6 b C b⊥
12	9, 11	fh3 471	. . . . 5 (b ∪ ((a⊥ ∪ b) ∩ b⊥ )) = ((b ∪ (a⊥ ∪ b)) ∩ (b ∪ b⊥ ))
13		or12 80	. . . . . . . 8 (b ∪ (a⊥ ∪ b)) = (a⊥ ∪ (b ∪ b))
14		oridm 110	. . . . . . . . 9 (b ∪ b) = b
15	14	lor 70	. . . . . . . 8 (a⊥ ∪ (b ∪ b)) = (a⊥ ∪ b)
16	13, 15	ax-r2 36	. . . . . . 7 (b ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
17		df-t 41	. . . . . . . 8 1 = (b ∪ b⊥ )
18	17	ax-r1 35	. . . . . . 7 (b ∪ b⊥ ) = 1
19	16, 18	2an 79	. . . . . 6 ((b ∪ (a⊥ ∪ b)) ∩ (b ∪ b⊥ )) = ((a⊥ ∪ b) ∩ 1)
20		an1 106	. . . . . 6 ((a⊥ ∪ b) ∩ 1) = (a⊥ ∪ b)
21	19, 20	ax-r2 36	. . . . 5 ((b ∪ (a⊥ ∪ b)) ∩ (b ∪ b⊥ )) = (a⊥ ∪ b)
22	12, 21	ax-r2 36	. . . 4 (b ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
23	8, 22	ax-r2 36	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
24	3, 23	ax-r2 36	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ b) = (a⊥ ∪ b)
25	2, 24	ax-r2 36	1 ((a →4 b) ∪ b) = (a⊥ ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →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:  u4lemnanb  658