Theorem u3lem3 751

Description: Lemma for unified implication study. (Contributed by NM, 17-Dec-1997.)

Assertion

Ref	Expression
u3lem3	(a →3 (b →3 a)) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Proof of Theorem u3lem3

Step	Hyp	Ref	Expression
1		df-i3 46	. 2 (a →3 (b →3 a)) = (((a⊥ ∩ (b →3 a)) ∪ (a⊥ ∩ (b →3 a)⊥ )) ∪ (a ∩ (a⊥ ∪ (b →3 a))))
2		ancom 74	. . . . . . 7 (a⊥ ∩ (b →3 a)) = ((b →3 a) ∩ a⊥ )
3		u3lemanb 617	. . . . . . 7 ((b →3 a) ∩ a⊥ ) = (b⊥ ∩ a⊥ )
4	2, 3	ax-r2 36	. . . . . 6 (a⊥ ∩ (b →3 a)) = (b⊥ ∩ a⊥ )
5		ancom 74	. . . . . . 7 (a⊥ ∩ (b →3 a)⊥ ) = ((b →3 a)⊥ ∩ a⊥ )
6		u3lemnanb 657	. . . . . . 7 ((b →3 a)⊥ ∩ a⊥ ) = (b ∩ a⊥ )
7	5, 6	ax-r2 36	. . . . . 6 (a⊥ ∩ (b →3 a)⊥ ) = (b ∩ a⊥ )
8	4, 7	2or 72	. . . . 5 ((a⊥ ∩ (b →3 a)) ∪ (a⊥ ∩ (b →3 a)⊥ )) = ((b⊥ ∩ a⊥ ) ∪ (b ∩ a⊥ ))
9		ancom 74	. . . . . . 7 (b⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
10		ancom 74	. . . . . . 7 (b ∩ a⊥ ) = (a⊥ ∩ b)
11	9, 10	2or 72	. . . . . 6 ((b⊥ ∩ a⊥ ) ∪ (b ∩ a⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b))
12		ax-a2 31	. . . . . 6 ((a⊥ ∩ b⊥ ) ∪ (a⊥ ∩ b)) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
13	11, 12	ax-r2 36	. . . . 5 ((b⊥ ∩ a⊥ ) ∪ (b ∩ a⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
14	8, 13	ax-r2 36	. . . 4 ((a⊥ ∩ (b →3 a)) ∪ (a⊥ ∩ (b →3 a)⊥ )) = ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ ))
15		ax-a2 31	. . . . . . 7 (a⊥ ∪ (b →3 a)) = ((b →3 a) ∪ a⊥ )
16		u3lemonb 637	. . . . . . 7 ((b →3 a) ∪ a⊥ ) = 1
17	15, 16	ax-r2 36	. . . . . 6 (a⊥ ∪ (b →3 a)) = 1
18	17	lan 77	. . . . 5 (a ∩ (a⊥ ∪ (b →3 a))) = (a ∩ 1)
19		an1 106	. . . . 5 (a ∩ 1) = a
20	18, 19	ax-r2 36	. . . 4 (a ∩ (a⊥ ∪ (b →3 a))) = a
21	14, 20	2or 72	. . 3 (((a⊥ ∩ (b →3 a)) ∪ (a⊥ ∩ (b →3 a)⊥ )) ∪ (a ∩ (a⊥ ∪ (b →3 a)))) = (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a)
22		ax-a2 31	. . 3 (((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )) ∪ a) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
23	21, 22	ax-r2 36	. 2 (((a⊥ ∩ (b →3 a)) ∪ (a⊥ ∩ (b →3 a)⊥ )) ∪ (a ∩ (a⊥ ∪ (b →3 a)))) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))
24	1, 23	ax-r2 36	1 (a →3 (b →3 a)) = (a ∪ ((a⊥ ∩ b) ∪ (a⊥ ∩ b⊥ )))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₃ wi3 14

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-i3 46  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  u3lem3n  754  u3lem14a  791