Theorem wql2lem2 289

Description: Lemma for →₂ WQL axiom. (Contributed by NM, 6-Dec-1998.)

Hypothesis

Ref	Expression
wql2lem2.1	((a ∪ c) →2 (b ∪ c)) = 1

Assertion

Ref	Expression
wql2lem2	((a ∪ (b ∪ c))⊥ ∪ (b ∪ c)) = 1

Proof of Theorem wql2lem2

Step	Hyp	Ref	Expression
1		df-i2 45	. . . 4 ((a ∪ c) →2 (b ∪ c)) = ((b ∪ c) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ ))
2		anor3 90	. . . . . 6 ((a ∪ c)⊥ ∩ (b ∪ c)⊥ ) = ((a ∪ c) ∪ (b ∪ c))⊥
3		ax-a3 32	. . . . . . . . . 10 ((a ∪ b) ∪ c) = (a ∪ (b ∪ c))
4	3	ax-r1 35	. . . . . . . . 9 (a ∪ (b ∪ c)) = ((a ∪ b) ∪ c)
5		orordir 113	. . . . . . . . 9 ((a ∪ b) ∪ c) = ((a ∪ c) ∪ (b ∪ c))
6	4, 5	ax-r2 36	. . . . . . . 8 (a ∪ (b ∪ c)) = ((a ∪ c) ∪ (b ∪ c))
7	6	ax-r4 37	. . . . . . 7 (a ∪ (b ∪ c))⊥ = ((a ∪ c) ∪ (b ∪ c))⊥
8	7	ax-r1 35	. . . . . 6 ((a ∪ c) ∪ (b ∪ c))⊥ = (a ∪ (b ∪ c))⊥
9	2, 8	ax-r2 36	. . . . 5 ((a ∪ c)⊥ ∩ (b ∪ c)⊥ ) = (a ∪ (b ∪ c))⊥
10	9	lor 70	. . . 4 ((b ∪ c) ∪ ((a ∪ c)⊥ ∩ (b ∪ c)⊥ )) = ((b ∪ c) ∪ (a ∪ (b ∪ c))⊥ )
11		ax-a2 31	. . . 4 ((b ∪ c) ∪ (a ∪ (b ∪ c))⊥ ) = ((a ∪ (b ∪ c))⊥ ∪ (b ∪ c))
12	1, 10, 11	3tr 65	. . 3 ((a ∪ c) →2 (b ∪ c)) = ((a ∪ (b ∪ c))⊥ ∪ (b ∪ c))
13	12	ax-r1 35	. 2 ((a ∪ (b ∪ c))⊥ ∪ (b ∪ c)) = ((a ∪ c) →2 (b ∪ c))
14		wql2lem2.1	. 2 ((a ∪ c) →2 (b ∪ c)) = 1
15	13, 14	ax-r2 36	1 ((a ∪ (b ∪ c))⊥ ∪ (b ∪ c)) = 1

Syntax hints:   = wb 1  ^⊥ wn 4   ∪ wo 6   ∩ wa 7  1wt 8   →₂ wi2 13

This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38

This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-i2 45

This theorem is referenced by:  wql2lem4  291