Theorem u2lem3 732

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u2lem3	(a →2 (b →2 a)) = 1

Proof of Theorem u2lem3

Step	Hyp	Ref	Expression
1		df-i2 44	. 2 (a →2 (b →2 a)) = ((b →2 a) ∪ (a⊥ ∩ (b →2 a)⊥ ))
2		u2lemc1 663	. . . . 5 a C (b →2 a)
3	2	comcom3 436	. . . 4 a⊥ C (b →2 a)
4	2	comcom4 437	. . . 4 a⊥ C (b →2 a)⊥
5	3, 4	fh4 454	. . 3 ((b →2 a) ∪ (a⊥ ∩ (b →2 a)⊥ )) = (((b →2 a) ∪ a⊥ ) ∩ ((b →2 a) ∪ (b →2 a)⊥ ))
6		u2lemonb 618	. . . . 5 ((b →2 a) ∪ a⊥ ) = 1
7		df-t 40	. . . . . 6 1 = ((b →2 a) ∪ (b →2 a)⊥ )
8	7	ax-r1 34	. . . . 5 ((b →2 a) ∪ (b →2 a)⊥ ) = 1
9	6, 8	2an 72	. . . 4 (((b →2 a) ∪ a⊥ ) ∩ ((b →2 a) ∪ (b →2 a)⊥ )) = (1 ∩ 1)
10		an1 98	. . . 4 (1 ∩ 1) = 1
11	9, 10	ax-r2 35	. . 3 (((b →2 a) ∪ a⊥ ) ∩ ((b →2 a) ∪ (b →2 a)⊥ )) = 1
12	5, 11	ax-r2 35	. 2 ((b →2 a) ∪ (a⊥ ∩ (b →2 a)⊥ )) = 1
13	1, 12	ax-r2 35	1 (a →2 (b →2 a)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₂ wi2 14

This theorem is referenced by:  imp3 823

This theorem was proved from axioms:  ax-a1 29  ax-a2 30  ax-a3 31  ax-a4 32  ax-a5 33  ax-r1 34  ax-r2 35  ax-r4 36  ax-r5 37  ax-r3 421

This theorem depends on definitions:  df-b 38  df-a 39  df-t 40  df-f 41  df-i2 44  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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