Theorem u2lem7n 775

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

Assertion

Ref	Expression
u2lem7n	(a →2 (a⊥ →2 b))⊥ = (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ b⊥ )

Proof of Theorem u2lem7n

Step	Hyp	Ref	Expression
1		u2lem7 773	. . 3 (a →2 (a⊥ →2 b)) = (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b)
2		ax-a2 31	. . . . . . 7 ((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) = ((a⊥ ∩ b⊥ ) ∪ (a ∩ b⊥ ))
3		anor3 90	. . . . . . . 8 (a⊥ ∩ b⊥ ) = (a ∪ b)⊥
4		anor1 88	. . . . . . . 8 (a ∩ b⊥ ) = (a⊥ ∪ b)⊥
5	3, 4	2or 72	. . . . . . 7 ((a⊥ ∩ b⊥ ) ∪ (a ∩ b⊥ )) = ((a ∪ b)⊥ ∪ (a⊥ ∪ b)⊥ )
6	2, 5	ax-r2 36	. . . . . 6 ((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) = ((a ∪ b)⊥ ∪ (a⊥ ∪ b)⊥ )
7		oran3 93	. . . . . 6 ((a ∪ b)⊥ ∪ (a⊥ ∪ b)⊥ ) = ((a ∪ b) ∩ (a⊥ ∪ b))⊥
8	6, 7	ax-r2 36	. . . . 5 ((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) = ((a ∪ b) ∩ (a⊥ ∪ b))⊥
9	8	ax-r5 38	. . . 4 (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b) = (((a ∪ b) ∩ (a⊥ ∪ b))⊥ ∪ b)
10		oran2 92	. . . 4 (((a ∪ b) ∩ (a⊥ ∪ b))⊥ ∪ b) = (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ b⊥ )⊥
11	9, 10	ax-r2 36	. . 3 (((a ∩ b⊥ ) ∪ (a⊥ ∩ b⊥ )) ∪ b) = (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ b⊥ )⊥
12	1, 11	ax-r2 36	. 2 (a →2 (a⊥ →2 b)) = (((a ∪ b) ∩ (a⊥ ∪ b)) ∩ b⊥ )⊥
13	12	con2 67	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:  u2lem8  782