Theorem u2lem3 750

Description: Lemma for unified implication study. (Contributed by NM, 17-Dec-1997.)

Assertion

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

Proof of Theorem u2lem3

Step	Hyp	Ref	Expression
1		df-i2 45	. 2 (a →2 (b →2 a)) = ((b →2 a) ∪ (a⊥ ∩ (b →2 a)⊥ ))
2		u2lemc1 681	. . . . 5 a C (b →2 a)
3	2	comcom3 454	. . . 4 a⊥ C (b →2 a)
4	2	comcom4 455	. . . 4 a⊥ C (b →2 a)⊥
5	3, 4	fh4 472	. . 3 ((b →2 a) ∪ (a⊥ ∩ (b →2 a)⊥ )) = (((b →2 a) ∪ a⊥ ) ∩ ((b →2 a) ∪ (b →2 a)⊥ ))
6		u2lemonb 636	. . . . 5 ((b →2 a) ∪ a⊥ ) = 1
7		df-t 41	. . . . . 6 1 = ((b →2 a) ∪ (b →2 a)⊥ )
8	7	ax-r1 35	. . . . 5 ((b →2 a) ∪ (b →2 a)⊥ ) = 1
9	6, 8	2an 79	. . . 4 (((b →2 a) ∪ a⊥ ) ∩ ((b →2 a) ∪ (b →2 a)⊥ )) = (1 ∩ 1)
10		an1 106	. . . 4 (1 ∩ 1) = 1
11	9, 10	ax-r2 36	. . 3 (((b →2 a) ∪ a⊥ ) ∩ ((b →2 a) ∪ (b →2 a)⊥ )) = 1
12	5, 11	ax-r2 36	. 2 ((b →2 a) ∪ (a⊥ ∩ (b →2 a)⊥ )) = 1
13	1, 12	ax-r2 36	1 (a →2 (b →2 a)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-r3 439

This theorem depends on definitions:  df-b 39  df-a 40  df-t 41  df-f 42  df-i2 45  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by:  imp3  841