Theorem u1lem2 744

Description: Lemma for unified implication study. (Contributed by NM, 16-Dec-1997.)

Assertion

Ref	Expression
u1lem2	(((a →1 b) →1 a) →1 a) = 1

Proof of Theorem u1lem2

Step	Hyp	Ref	Expression
1		df-i1 44	. 2 (((a →1 b) →1 a) →1 a) = (((a →1 b) →1 a)⊥ ∪ (((a →1 b) →1 a) ∩ a))
2		u1lem1n 739	. . . 4 ((a →1 b) →1 a)⊥ = a⊥
3		u1lem1 734	. . . . . 6 ((a →1 b) →1 a) = a
4	3	ran 78	. . . . 5 (((a →1 b) →1 a) ∩ a) = (a ∩ a)
5		anidm 111	. . . . 5 (a ∩ a) = a
6	4, 5	ax-r2 36	. . . 4 (((a →1 b) →1 a) ∩ a) = a
7	2, 6	2or 72	. . 3 (((a →1 b) →1 a)⊥ ∪ (((a →1 b) →1 a) ∩ a)) = (a⊥ ∪ a)
8		ax-a2 31	. . . 4 (a⊥ ∪ a) = (a ∪ a⊥ )
9		df-t 41	. . . . 5 1 = (a ∪ a⊥ )
10	9	ax-r1 35	. . . 4 (a ∪ a⊥ ) = 1
11	8, 10	ax-r2 36	. . 3 (a⊥ ∪ a) = 1
12	7, 11	ax-r2 36	. 2 (((a →1 b) →1 a)⊥ ∪ (((a →1 b) →1 a) ∩ a)) = 1
13	1, 12	ax-r2 36	1 (((a →1 b) →1 a) →1 a) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₁ wi1 12

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-i1 44  df-le1 130  df-le2 131  df-c1 132  df-c2 133

This theorem is referenced by: (None)