Theorem u2lem5 762

Description: Lemma for unified implication study. (Contributed by NM, 20-Dec-1997.)

Assertion

Ref	Expression
u2lem5	(a →2 (a →2 b)) = (a →2 b)

Proof of Theorem u2lem5

Step	Hyp	Ref	Expression
1		df-i2 45	. 2 (a →2 (a →2 b)) = ((a →2 b) ∪ (a⊥ ∩ (a →2 b)⊥ ))
2		ancom 74	. . . . 5 (a⊥ ∩ (a →2 b)⊥ ) = ((a →2 b)⊥ ∩ a⊥ )
3		u2lemnana 646	. . . . 5 ((a →2 b)⊥ ∩ a⊥ ) = 0
4	2, 3	ax-r2 36	. . . 4 (a⊥ ∩ (a →2 b)⊥ ) = 0
5	4	lor 70	. . 3 ((a →2 b) ∪ (a⊥ ∩ (a →2 b)⊥ )) = ((a →2 b) ∪ 0)
6		or0 102	. . 3 ((a →2 b) ∪ 0) = (a →2 b)
7	5, 6	ax-r2 36	. 2 ((a →2 b) ∪ (a⊥ ∩ (a →2 b)⊥ )) = (a →2 b)
8	1, 7	ax-r2 36	1 (a →2 (a →2 b)) = (a →2 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 9   →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-t 41  df-f 42  df-i2 45

This theorem is referenced by: (None)