Theorem i1orni1 847

Description: Complemented antecedent lemma. (Contributed by NM, 6-Aug-2001.)

Assertion

Ref	Expression
i1orni1	((a →1 b) ∪ (a⊥ →1 b)) = 1

Proof of Theorem i1orni1

Step	Hyp	Ref	Expression
1		df-i1 44	. . . 4 (a⊥ →1 b) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
2		ax-a1 30	. . . . . 6 a = a⊥ ⊥
3	2	ax-r5 38	. . . . 5 (a ∪ (a⊥ ∩ b)) = (a⊥ ⊥ ∪ (a⊥ ∩ b))
4	3	ax-r1 35	. . . 4 (a⊥ ⊥ ∪ (a⊥ ∩ b)) = (a ∪ (a⊥ ∩ b))
5	1, 4	ax-r2 36	. . 3 (a⊥ →1 b) = (a ∪ (a⊥ ∩ b))
6	5	lor 70	. 2 ((a →1 b) ∪ (a⊥ →1 b)) = ((a →1 b) ∪ (a ∪ (a⊥ ∩ b)))
7		orordi 112	. . 3 ((a →1 b) ∪ (a ∪ (a⊥ ∩ b))) = (((a →1 b) ∪ a) ∪ ((a →1 b) ∪ (a⊥ ∩ b)))
8		u1lemoa 620	. . . . 5 ((a →1 b) ∪ a) = 1
9	8	ax-r5 38	. . . 4 (((a →1 b) ∪ a) ∪ ((a →1 b) ∪ (a⊥ ∩ b))) = (1 ∪ ((a →1 b) ∪ (a⊥ ∩ b)))
10		or1r 105	. . . 4 (1 ∪ ((a →1 b) ∪ (a⊥ ∩ b))) = 1
11	9, 10	ax-r2 36	. . 3 (((a →1 b) ∪ a) ∪ ((a →1 b) ∪ (a⊥ ∩ b))) = 1
12	7, 11	ax-r2 36	. 2 ((a →1 b) ∪ (a ∪ (a⊥ ∩ b))) = 1
13	6, 12	ax-r2 36	1 ((a →1 b) ∪ (a⊥ →1 b)) = 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

This theorem depends on definitions:  df-t 41  df-f 42  df-i1 44

This theorem is referenced by:  negantlem2  849