Theorem u4lemob 615

Description: Lemma for non-tollens implication study.

Assertion

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

Proof of Theorem u4lemob

Step	Hyp	Ref	Expression
1		df-i4 46	. . 3 (a →4 b) = (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
2	1	ax-r5 37	. 2 ((a →4 b) ∪ b) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ b)
3		or32 75	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ b) = ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ ))
4		lear 153	. . . . . . 7 (a ∩ b) ≤ b
5		lear 153	. . . . . . 7 (a⊥ ∩ b) ≤ b
6	4, 5	lel2or 162	. . . . . 6 ((a ∩ b) ∪ (a⊥ ∩ b)) ≤ b
7	6	df-le2 123	. . . . 5 (((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) = b
8	7	ax-r5 37	. . . 4 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (b ∪ ((a⊥ ∪ b) ∩ b⊥ ))
9		comorr2 445	. . . . . 6 b C (a⊥ ∪ b)
10		comid 179	. . . . . . 7 b C b
11	10	comcom2 175	. . . . . 6 b C b⊥
12	9, 11	fh3 453	. . . . 5 (b ∪ ((a⊥ ∪ b) ∩ b⊥ )) = ((b ∪ (a⊥ ∪ b)) ∩ (b ∪ b⊥ ))
13		or12 73	. . . . . . . 8 (b ∪ (a⊥ ∪ b)) = (a⊥ ∪ (b ∪ b))
14		oridm 102	. . . . . . . . 9 (b ∪ b) = b
15	14	lor 66	. . . . . . . 8 (a⊥ ∪ (b ∪ b)) = (a⊥ ∪ b)
16	13, 15	ax-r2 35	. . . . . . 7 (b ∪ (a⊥ ∪ b)) = (a⊥ ∪ b)
17		df-t 40	. . . . . . . 8 1 = (b ∪ b⊥ )
18	17	ax-r1 34	. . . . . . 7 (b ∪ b⊥ ) = 1
19	16, 18	2an 72	. . . . . 6 ((b ∪ (a⊥ ∪ b)) ∩ (b ∪ b⊥ )) = ((a⊥ ∪ b) ∩ 1)
20		an1 98	. . . . . 6 ((a⊥ ∪ b) ∩ 1) = (a⊥ ∪ b)
21	19, 20	ax-r2 35	. . . . 5 ((b ∪ (a⊥ ∪ b)) ∩ (b ∪ b⊥ )) = (a⊥ ∪ b)
22	12, 21	ax-r2 35	. . . 4 (b ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
23	8, 22	ax-r2 35	. . 3 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ b) ∪ ((a⊥ ∪ b) ∩ b⊥ )) = (a⊥ ∪ b)
24	3, 23	ax-r2 35	. 2 ((((a ∩ b) ∪ (a⊥ ∩ b)) ∪ ((a⊥ ∪ b) ∩ b⊥ )) ∪ b) = (a⊥ ∪ b)
25	2, 24	ax-r2 35	1 ((a →4 b) ∪ b) = (a⊥ ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₄ wi4 16

This theorem is referenced by:  u4lemnanb 640

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i4 46  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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