Theorem thm3.8i1lem 1080

Description: Lemma intended for ~ thm3.8i1 . (Contributed by Roy F. Longton, 30-Jun-2005.) (Revised by Roy F. Longton, 31-Mar-2011.)

Assertion

Ref	Expression
thm3.8i1lem	(a ≡1 b) = ((b →0 a) ∩ (a →1 b))

Proof of Theorem thm3.8i1lem

Step	Hyp	Ref	Expression
1		ax-a2 31	. . 3 (a ∪ b⊥ ) = (b⊥ ∪ a)
2	1	ran 78	. 2 ((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b))) = ((b⊥ ∪ a) ∩ (a⊥ ∪ (a ∩ b)))
3		df-id1 50	. 2 (a ≡1 b) = ((a ∪ b⊥ ) ∩ (a⊥ ∪ (a ∩ b)))
4		df-i0 43	. . 3 (b →0 a) = (b⊥ ∪ a)
5		df-i1 44	. . 3 (a →1 b) = (a⊥ ∪ (a ∩ b))
6	4, 5	2an 79	. 2 ((b →0 a) ∩ (a →1 b)) = ((b⊥ ∪ a) ∩ (a⊥ ∪ (a ∩ b)))
7	2, 3, 6	3tr1 63	1 (a ≡1 b) = ((b →0 a) ∩ (a →1 b))

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7   →₀ wi0 11   →₁ wi1 12   ≡₁ wid1 18

This theorem was proved from axioms:  ax-a2 31  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-i0 43  df-i1 44  df-id1 50

This theorem is referenced by: (None)