Theorem u2lem5 744

Description: Lemma for unified implication study.

Assertion

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

Proof of Theorem u2lem5

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 (a →2 (a →2 b)) = ((a →2 b) ∪ (a⊥ ∩ (a →2 b)⊥ ))
2		ancom 68	. . . . 5 (a⊥ ∩ (a →2 b)⊥ ) = ((a →2 b)⊥ ∩ a⊥ )
3		u2lemnana 628	. . . . 5 ((a →2 b)⊥ ∩ a⊥ ) = 0
4	2, 3	ax-r2 35	. . . 4 (a⊥ ∩ (a →2 b)⊥ ) = 0
5	4	lor 66	. . 3 ((a →2 b) ∪ (a⊥ ∩ (a →2 b)⊥ )) = ((a →2 b) ∪ 0)
6		or0 94	. . 3 ((a →2 b) ∪ 0) = (a →2 b)
7	5, 6	ax-r2 35	. 2 ((a →2 b) ∪ (a⊥ ∩ (a →2 b)⊥ )) = (a →2 b)
8	1, 7	ax-r2 35	1 (a →2 (a →2 b)) = (a →2 b)

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  0wf 10   →₂ wi2 14

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

This theorem depends on definitions:  df-a 39  df-t 40  df-f 41  df-i2 44

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