Theorem u1lem2 726

Description: Lemma for unified implication study.

Assertion

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

Proof of Theorem u1lem2

Step	Hyp	Ref	Expression
1		df-i1 43	. 2 (((a →1 b) →1 a) →1 a) = (((a →1 b) →1 a)⊥ ∪ (((a →1 b) →1 a) ∩ a))
2		u1lem1n 721	. . . 4 ((a →1 b) →1 a)⊥ = a⊥
3		u1lem1 716	. . . . . 6 ((a →1 b) →1 a) = a
4	3	ran 71	. . . . 5 (((a →1 b) →1 a) ∩ a) = (a ∩ a)
5		anidm 103	. . . . 5 (a ∩ a) = a
6	4, 5	ax-r2 35	. . . 4 (((a →1 b) →1 a) ∩ a) = a
7	2, 6	2or 67	. . 3 (((a →1 b) →1 a)⊥ ∪ (((a →1 b) →1 a) ∩ a)) = (a⊥ ∪ a)
8		ax-a2 30	. . . 4 (a⊥ ∪ a) = (a ∪ a⊥ )
9		df-t 40	. . . . 5 1 = (a ∪ a⊥ )
10	9	ax-r1 34	. . . 4 (a ∪ a⊥ ) = 1
11	8, 10	ax-r2 35	. . 3 (a⊥ ∪ a) = 1
12	7, 11	ax-r2 35	. 2 (((a →1 b) →1 a)⊥ ∪ (((a →1 b) →1 a) ∩ a)) = 1
13	1, 12	ax-r2 35	1 (((a →1 b) →1 a) →1 a) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 9   →₁ wi1 13

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-i1 43  df-le1 122  df-le2 123  df-c1 124  df-c2 125

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