Theorem u2lem7 773

Description: Lemma for unified implication study. (Contributed by NM, 24-Dec-1997.)

Assertion

Ref	Expression
u2lem7	(a →2 (a⊥ →2 b)) = (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b)

Proof of Theorem u2lem7

Step	Hyp	Ref	Expression
1		df-i2 45	. 2 (a →2 (a⊥ →2 b)) = ((a⊥ →2 b) ∪ (a⊥ ∩ (a⊥ →2 b)⊥ ))
2		df-i2 45	. . . . 5 (a⊥ →2 b) = (b ∪ (a⊥ ⊥ ∩ b⊥ ))
3		ax-a1 30	. . . . . . . 8 a = a⊥ ⊥
4	3	ax-r1 35	. . . . . . 7 a⊥ ⊥ = a
5	4	ran 78	. . . . . 6 (a⊥ ⊥ ∩ b⊥ ) = (a ∩ b⊥ )
6	5	lor 70	. . . . 5 (b ∪ (a⊥ ⊥ ∩ b⊥ )) = (b ∪ (a ∩ b⊥ ))
7	2, 6	ax-r2 36	. . . 4 (a⊥ →2 b) = (b ∪ (a ∩ b⊥ ))
8		ancom 74	. . . . 5 (a⊥ ∩ (a⊥ →2 b)⊥ ) = ((a⊥ →2 b)⊥ ∩ a⊥ )
9		u2lemnaa 641	. . . . 5 ((a⊥ →2 b)⊥ ∩ a⊥ ) = (a⊥ ∩ b⊥ )
10	8, 9	ax-r2 36	. . . 4 (a⊥ ∩ (a⊥ →2 b)⊥ ) = (a⊥ ∩ b⊥ )
11	7, 10	2or 72	. . 3 ((a⊥ →2 b) ∪ (a⊥ ∩ (a⊥ →2 b)⊥ )) = ((b ∪ (a ∩ b⊥ )) ∪ (a⊥ ∩ b⊥ ))
12		ax-a3 32	. . . 4 ((b ∪ (a ∩ b⊥ )) ∪ (a⊥ ∩ b⊥ )) = (b ∪ ((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )))
13		ax-a2 31	. . . 4 (b ∪ ((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ ))) = (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b)
14	12, 13	ax-r2 36	. . 3 ((b ∪ (a ∩ b⊥ )) ∪ (a⊥ ∩ b⊥ )) = (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b)
15	11, 14	ax-r2 36	. 2 ((a⊥ →2 b) ∪ (a⊥ ∩ (a⊥ →2 b)⊥ )) = (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b)
16	1, 15	ax-r2 36	1 (a →2 (a⊥ →2 b)) = (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i2 45  df-le1 130  df-le2 131

This theorem is referenced by:  u2lem7n  775  u2lem8  782