Theorem u1lem7 754

Description: Lemma for unified implication study.

Assertion

Ref	Expression
u1lem7	(a →1 (a⊥ →1 b)) = 1

Proof of Theorem u1lem7

Step	Hyp	Ref	Expression
1		df-i1 43	. 2 (a →1 (a⊥ →1 b)) = (a⊥ ∪ (a ∩ (a⊥ →1 b)))
2		ax-a1 29	. . . . . 6 a = a⊥ ⊥
3	2	ran 71	. . . . 5 (a ∩ (a⊥ →1 b)) = (a⊥ ⊥ ∩ (a⊥ →1 b))
4		ancom 68	. . . . . 6 (a⊥ ⊥ ∩ (a⊥ →1 b)) = ((a⊥ →1 b) ∩ a⊥ ⊥ )
5		u1lemana 587	. . . . . 6 ((a⊥ →1 b) ∩ a⊥ ⊥ ) = a⊥ ⊥
6	4, 5	ax-r2 35	. . . . 5 (a⊥ ⊥ ∩ (a⊥ →1 b)) = a⊥ ⊥
7	3, 6	ax-r2 35	. . . 4 (a ∩ (a⊥ →1 b)) = a⊥ ⊥
8	7	lor 66	. . 3 (a⊥ ∪ (a ∩ (a⊥ →1 b))) = (a⊥ ∪ a⊥ ⊥ )
9		df-t 40	. . . 4 1 = (a⊥ ∪ a⊥ ⊥ )
10	9	ax-r1 34	. . 3 (a⊥ ∪ a⊥ ⊥ ) = 1
11	8, 10	ax-r2 35	. 2 (a⊥ ∪ (a ∩ (a⊥ →1 b))) = 1
12	1, 11	ax-r2 35	1 (a →1 (a⊥ →1 b)) = 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-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-i1 43

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